Ice-stream margins in West Antarctica form from the transition between fast-flowing and stagnant ice. Observations and modelling studies suggest that intense lateral deformation of ice at the margins generates internal meltwater within the ice sheet. Röthlisberger channels (R-channels) may form in which water flows in the downstream direction along the margin. Standard theory argues that the steady-state channel diameter comes from the balance of in-plane creeping and melting rates. However, at the margin, large shear strains acting in the direction parallel to their cross-sectional axis (in the anti-plane direction) can develop. In view of understanding how R-channels evolve within ice streams, we construct a finite-element model (FEM) of an R-channel as considered by Nye (1953) with superimposed anti-plane shear stresses applied along the bed of the ice sheet on the boundary outside of the channel. This is similar to the model presented by Weertman (1972) to describe R-channels formed at the base of mountain glaciers. Weertman’s analysis suggested that the presence of large anti-plane shear stresses, as compared with in-plane stresses, makes the in-plane flow law effectively linear viscous (pseudo-Newtonian). Using our FEM, we examine the limitations of Weertman’s solution as well as when his linear viscous model may be adequate. Furthermore, we examine the effects of shear stresses acting transverse to the channel axis as a result of the interlocking of the bed and the ice. This, as pointed out by Weertman, will change the in-plane solution from being axisymmetric; ultimately making the assumption of a Nye solution near the bed invalid. In addition, we attempt to modify the basal boundary conditions to incorporate the hydrological model proposed by Perol and others (2014), where the basal shear stress varies as a function of the distance from the channel. Ultimately, this makes a comprehensive model for R-channels with anti-plane shear stress hypothesized to exist in ice-stream shear margins and mountain glaciers.