A single degree-of-freedom model of offshore structures with bilinear stiffness is studied for its dynamic response to a nonzero mean, oscillatory fluid flow. Of particular interest is the interaction between the bilinear stiffness characteristic and the asymmetric hydrodynamic drag force. The shooting method and the incremental harmonic balance method are used to determine the periodic steady state solutions. The stability of these solutions is examined by numerically computing the Floquet multipliers. The stability boundaries are also verified by comparison with the results obtained by direct numerical integration. The cases of stiffness ratio >1 and <1 are investigated. A large primary resonance, several subharmonic resonances as well as small superharmonic resonances are found to exist. At higher wave frequencies, multiple stable steady state solutions can coexist. The presence of nonzero current gives rise to a significant qualitative change in the response. Hysteresis and asymmetry in the subharmonic solutions appear when the asymmetric drag force is introduced. For stiffness ratios >1, successive period-doubling bifurcations result for some odd-order subharmonic responses. Further increases in the stiffness ratio can result in a "crisis" in that higher-order subharmonics suddenly disappear and for every initial condition, the solutions, after long transients, converge to solutions with the exciting wave frequency. For stiffness ratios <1, large subharmonic resonances are observed. They exist over a wide range in the wave frequency. Also, successive period-doubling bifurcations as well as the crisis are not exhibited over the parameter range investigated.