An adaptation of the space-harmonics series is presented for analyzing periodic structures with finite boundaries. When the boundaries are simply supported, the expansion for the response takes the form of a sine series. The selection of the arguments of the sine functions is, however, less straightforward than in the space harmonics series for infinite structures, due to the requirement that the functions be linearly independent. The applicability of the method is examined for simply supported beams and plates with properties that vary periodically in space. It is shown that when each periodic unit is symmetric about its mid-point, the sine expansion developed here produces stiffness and mass matrices which are block-diagonal.