A nonconforming finite element method (FEM) is proposed and analyzed for the clamped thin elastic Kirchhoff plate unilaterally constrained by an elastic obstacle. The discrete scheme is constructed by using the strongly discontinuous Bergan's energy-orthogonal plate element, which has simple degrees of freedom and about 25 percent fewer global dimension than that of the famous triangular Morley element. A novel error analysis is presented to overcome the difficulties caused by the strong discontinuity and derive the optimal estimate. Numerical experiments are carried out to verify the theoretical analysis.