Scalar damage models are very often implemented in computational analyses in order to predict the response and failure modes of concrete and reinforced concrete structures. In most situations, however, damage is not isotropic but has preferential directions. Therefore, there have been many questions about the pertinence and range of applicability of isotropic, scalar, damage models for describing a degradation process which is strongly geometrically oriented. In order to assess what are the limitations of such a simplifying assumption, a comparative study is presented. The constitutive relations used for this purpose derive from the same class of models with a gradual enhancement of the description of damage. The scalar damage model is compared to another model where damage-induced orthotropy is described, with the possibility of rotation of the principle axes of orthotropy. Both models incorporate crack closure e!ects and a plasticity damage coupling. Structural analyses on bending beams, compression-shear and tension-shear concrete panels are presented. Although it may appear to be simplistic, the scalar damage model provides accurate predictions when failure is mainly due to uniaxial extension. Crack closure introduces an additional anisotropy which is important in compression-shear problems. Finally, damage-induced anisotropy seems important when failure is due to multiaxial extensions, such as in shear-tension problems.