Conical structures are often used to ameliorate the action of floating ice. There is a long history of using cone-shaped bridge piers. Moreover, many recent designs adopt downward-breaking cones to further reduce ice forces. The present work aims to clarify aspects of ice interaction with cones that are poorly understood. This work builds on a previous study that employed a numerical model of ice dynamics in order to predict ice failure patterns and forces on a conical structure. Performance of the model was validated against available ice basin tests, and the study examined a test case representing a pier of the Confederation Bridge, Canada. The present tests examine the roles of ice-structure friction and shape of the cone (upward- or downward-breaking) in more detail. Additionally, the effects of ice thickness and embedded consolidated ridges are examined. The results reveal trends of the dependence of the modes of ice failure and the resulting forces on ice-structure friction, slope and the type of the cone (upward- or downward-breaking). For upward-breaking cones, ice-structure friction proved to be more significant for gentler slopes than for steeper ones. For those cones, ice failure appeared to correspond to three-dimensional buckling. Downward-breaking cones displayed a different behavior with downwards bending failure. The value of the ice-structure friction coefficient had a clear influence on the forces for all downward-breaking cases. The results also support the idea that downward-breaking cones correspond to lower horizontal forces than those acting on upward-breaking cones. Concerning ice sheet thickness, peak horizontal ice forces show a near linear dependence on the thickness. For large consolidated ridges, forces increased substantially. The numerical model results are compared with the elastic beam bending approach given in the ISO 19906 Arctic Offshore Structures Standard for ice loading on a sloped structure. In order to use the equations in ISO 19906, assumptions must be made on the maximum accumulated rubble height and other parameters, which affect the calculated load. The peak loads from the model and ISO approach are similar for level ice interactions, although the ISO approach is more sensitive to ice-structure friction. Also, the ISO approach may overestimate the load generated by some ridge geometries.