Additively manufactured components include a support structure and a component body integrally formed with the support structure using an additive-manufacturing process. The support structure includes an array of support members and at least some of the support members have a conduction gate such that the array of support members includes conduction gates distributed at a multitude of locations along the vertical axis of the respective support members. Methods of securing a body of a component to a build platform during additive manufacturing include forming a support structure and a component body integrally with the support structure using the additive-manufacturing process, in which the support structure includes an array of support members and at least some of the support members have a conduction gate such that the array of support members includes conduction gates distributed at a multitude of locations along the vertical axis of the respective support members. The support members that include a conduction gate have a first cross-sectional width and the conduction gate has a second cross-sectional width narrower than the first cross-sectional width.