  We construct moduli stacks of stable sheaves for surfaces fibered over marked
nodal curves by using expanded degenerations. These moduli stacks carry a
virtual class and therefore give rise to enumerative invariants. In the case of
a surface with two irreducible components glued along a smooth divisor, we
prove a degeneration formula that relates the moduli space associated to the
surface with the relative spaces associated to the two components. For a smooth
surface and no markings, our notion of stability agrees with slope stability
with respect to a suitable choice of polarization. We apply our results to
compute elliptic genera of moduli spaces of stable sheaves on some elliptic
surfaces.
