Textile fabrics are highly anisotropic, so that their mechanical properties including strengths are a function of direction. An extreme case is when a woven fabric sample is cut in such a way where the bias angle and hence the tension loading direction is around 45° relative to the principal directions. Then, once loaded, no yarn in the sample is held at both ends, so the yarns have to build up their internal tension entirely via yarn–yarn friction at the interlacing points. The overall fabric strength in such a sample is a result of contributions from the yarns being pulled out and those broken during the process, and thus becomes a function of the bias direction angle θ, sample width W and length L, along with other factors known to affect fabric strength tested in principal directions. Furthermore, in such a bias sample when the major parameters, e.g. the sample width W, change, not only the resultant strengths differ, but also the strength generating mechanisms (or failure types) vary. This is an interesting problem and is analysed in this study. More specifically, the issues examined in this paper include the exact mechanisms and details of how each interlacing point imparts the frictional constraint for a yarn to acquire tension to the level of its strength when both yarn ends were not actively held by the testing grips; the theoretical expression of the critical yarn length for a yarn to be able to break rather than be pulled out, as a function of the related factors; and the general relations between the tensile strength of such a bias sample and its structural properties. At the end, theoretical predictions are compared with our experimental data.