In most structural members with initial cracks (or initial notches), strength tends to decrease as the member size increases: a phenomenon known as size effect. In contrast to other structural materials (wood, steel etc.), concrete shows the size effect with or without initial crack. Published experimental results indicate that even a large concrete member without initial crack resists some stress, contrary to Bazant's size effect law. In this study, the size effect for concrete with dissimilar initial cracks is discussed on the basis of non-linear fracture mechanics, and prediction models are proposed for splitting tensile strength, shear strength and uniaxial compressive strength tests. The models are derived using regression analyses, with extensive experimental data relating to specimens of various sizes; it is felt that they perform much better than Bazant's model, particularly for very large specimens.