  We derive an off-shell formulation for the boundary Fayet-Iliopoulos (FI)
terms in locally supersymmetric U(1) gauge theory on 5D $S^1/Z_2$ orbifold.
Some physical consequences of such FI terms, e.g., the supersymmetry breaking
and the generation of 5D kink mass for hypermultiplet are studied within the
full supergravity framework. We especially find that the supersymmetry is
broken by the FI term without charged hypermultiplet for models giving an
AdS$_5$ geometry.
