In edge-notched tension specimens of unidirectional fiber-polymer composites, an initial transverse crack or notch might not propagate forward. Instead, sideways cracks parallel to fibers may grow axially from the initial crack tip. When the fibers are inclined from the specimen axis, a sideways crack propagates along the inclined fibers. Unless a potential sideways orthogonal cohesive crack is considered to exist a priori at the right location, the cohesive crack model cannot predict the sideways branching correctly. The crack band model can, because it involves a damage model with tensorial strength criterion, instead of scalar stress-separation law. We obtain approximate analytical solutions by the method of stress relief zones bordered by lines of slope k calibrated by J -integral. Our analysis, verified by finite element simulations, yields formulae for the structure size effect of orthogonal or inclined sideways cracks, taking into account the effect of the length of original transverse crack or notch, and the effect of the ratio of fracture energies G f and Gs of the forward and sideways cracks, respectively. The sideways cracks develop only if the ratioGs/G f is small enough, below a certain critical value (on-the-order of 10-1) depending on material parameters. Measuring the size effect of sideways cracks yields Gs and the corresponding fracture process zone size. The theory is shown tomatch Nairn's tests with varying notch depths. The results can be applied to all kinds of orthotropic composites.