The optimal design of composite structures requires progresses in the understanding of the mechanism of failure of fibrous composite materials. As regards the micromechanical structure of fibrous composites, the authors have already shed light on several not completely understood aspects of composites behaviour. In particular, the mechanism of failure of a fibrous composite under tension transverse to the fibres has recently been explained by the authors, establishing a relation between the observed failure and the predictions of initiation and extension of the damage. The objective of this paper is to contribute to find an explanation for the appearance of an angle of failure of about 53°, referenced by several authors, in the case of compression transverse to the fibres. The analysis is carried out by means of Boundary Elements, using the concepts derived from Interfacial Fracture Mechanics. The study developed shows that, following a scheme of analysis of the damage mechanism analogous to that employed for the tension case, the initial direction of failure predicted agrees with that observed in experiments.