We study the gravitational collapse of radiating stars in the context of the cosmic censorship conjecture. We consider a generalized Vaidya spacetime with three concentric regions. The local internal atmosphere is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the Schwarzschild exterior. We outline the general mathematical framework to study the conditions on the mass function so that future-directed nonspacelike geodesics can terminate at the singularity in the past. Mass functions for several equations of state are analyzed using this framework and it is shown that the collapse in each case terminates at a locally naked central singularity. We calculate the strength of these singularities to show that they are strong curvature singularities which implies that no extension of spacetime through them is possible.