Flapping flight and swimming in many organisms is accompanied by significant bending of flexible wings and fins. The instantaneous shape of wings and fins has, in turn, a profound effect on the fluid dynamic forces they can generate, with non-monotonic relationships between the pattern of deformation waves passing along the wing and the thrust developed. Many of these deformations arise, in part, from the passive mechanics of oscillating a flexible air- or hydrofoil. At the same time, however, their instantaneous shape may well emerge from details of the fluid loading. This issue—the extent to which there is feedback between the instantaneous wing shape and the fluid dynamic loading—is core to understanding flight control. We ask to what extent surface shape of wings and fins is controlled by structural mechanics versus fluid dynamic loading. To address this issue, we use a combination of computational and analytic methods to explore how bending stresses arising from inertial-elastic mechanisms compare to those stresses that arise from fluid pressure forces. Our analyses suggest that for certain combinations of wing stiffness, wing motions, and fluid density, fluid pressure stresses play a relatively minor role in determining wing shape. Nearly all of these combinations correspond to wings moving in air. The exciting feature provided by this analysis is that, for high Reynolds number motions where linear potential flow equations provide reasonable estimates of lift and thrust, we can finally examine how wing structure affects flight performance. Armed with this approach, we then show how modest levels of passive elasticity can affect thrust for a given level of energy input in the form of an inertial oscillation of a compliant foil.