Offshore drilling operations typically involve a wellhead/casing system connected to a floating vessel, with metocean forcing on the vessel translating to lateral movement at the wellhead. The conductor is the outer-most casing and must be checked against fatigue due to repeated cycling, which can create ‘hot-spots’ of high accumulated damage. Assessing this involves soil-structure interaction analysis, in which the soil-conductor lateral behaviour is modelled as non-linear springs called p-y curves. Current industry practice often only accounts for stiffness degradation due to cyclic loading, excluding load history effects and/or potential regain in stiffness resulting from pore-pressure dissipation after cycling. This paper compares results from centrifuge testing of a rigid length of conductor installed in reconstituted samples of carbonate silt and kaolin clay, subject to sequences of cyclic lateral displacement. The main aspects studied are the effect of previous cycling (load history), the effect of consolidation between episodes of undrained loading, and the impact of one-way vs two-way loading.