The radial structure of the dissipative trapped ion mode is studied in order to determine the effect of shear and the possibility of finding ranges of tokamak operation which are stable to this mode even in the presence of a temperature gradient. It is found that owing to the presence of increasing numbers of resonant untrapped ions near the mode rational surfaces, wherelqis an integer, the radial wavelengths are reduced and the modes are strongly Landau damped. These effects increase with increasing shear and with increasing values ofl, the toroidal mode number. Numerical estimates of the growth rates and marginal stability conditions are obtained from eigenmode solutions of the radial differential equation and from algebraic dispersion relations which simulate the effects of the radial structure. It is concluded that if(r /q )d q / d rexceeds one, then the temperature threshold for instability is raised by approximately a factor of two from that of the no shear case. It is also proposed that even in unstable regimes, the anomalous diffusion should be less than indicated by previous estimates, since both the growth rates and wavelengths are reduced.