If a rigid wheel or roller acts on an infinitely-long beam or a narrow (1-D) plate that is attached to a deformable foundation, point contact or a region of circular contact may occur. The following problems are analyzed: (1) an Euler-Bernoulli beam or an elastica attached to a Winkler foundation; (2) an Euler-Bernoulli beam or an elastica attached to a Pasternak foundation; (3) an Euler-Bernoulli beam attached to a foundation with two different stiffnesses; (4) point contact of a moving wheel with constant speed on an Euler-Bernoulli beam or an elastica attached to a Winkler or Pasternak foundation; (5) point contact of a 1-D plate on an elastomeric foundation; and (6) point contact of a 1-D plate attached to a combined foundation (including Winkler and elastomeric actions). The Euler-Bernoulli beam model tends to predict larger maximum deflections than the elastica model. For small speeds, the maximum deflection increases as the speed increases.