  We discuss a gauged $U(1)_R$ supergravity on five-dimensional (5D) orbifold
($S^1/Z_2$) in which both a $Z_2$-even U(1) gauge field and the $Z_2$-odd
graviphoton take part in the $U(1)_R$ gauging. Based on the off-shell
formulation of 5D supergravity, we analyze the structure of Fayet-Iliopoulos
(FI) terms allowed in such model. Introducing a $Z_2$-even $U(1)_R$ gauge field
accompanies new bulk and boundary FI terms in addition to the known integrable
boundary FI term which could be present in the absence of any gauged $U(1)_R$
symmetry. Some physical consequences of these new FI terms are examined.
