This invention provides a flooring apparatus comprising: (a) a flooring plate; (b) a plurality of columns extending from an underside of the flooring plate, wherein at least some of the columns deform when the flooring plate is subjected to a pressure equal to or greater than a critical buckling pressure; and (c) a matrix material in at least partial contact with the columns. The present disclosure includes many variations of such apparatus, such as various column designs, material selection criteria, and end-use considerations. This invention also provides a flooring system for mitigating injuries associated with falls, the system comprising a plurality of flooring apparatus each provided in accordance the present disclosure. The flooring system can include one or more types of modular tiles, or any other configuration (such as mats).