A combination of the Galerkin procedure and the method of multiple scales is used to analyze the nonlinear forced response of circular cylindrical shells in the presence of internal (autoparametric) resonances. If ω_f and a_f denote the frequency and amplitude of a flexural mode and ω_b and a_b denote the frequency and amplitude of the breathing mode, the steady-state response exhibits a saturation phenomenon when ω_b ≈ 2w_f if the shell is excited by a harmonic load having a frequency Ω near ω_b. As the amplitude f of the excitation increases from zero, a_b increases linearly whereas a_f remains zero until a threshold is reached. This threshold is a function of the damping coefficients and ω_b -2w_f. Beyond this threshold, a_b remains constant (i.e., the breathing mode saturates) and the extra energy spills over into the flexural mode. In other words, although the breathing mode is directly excited by the load, it absorbs a small amount of the input energy (responds with a small amplitude) and passes the rest of the input energy into the flexural mode (responds with a large amplitude).