  We use the Galaxies-Intergalactic Medium Interaction Calculation (GIMIC)
suite of cosmological hydrodynamical simulations to study the global structure
and kinematics of stellar spheroids of Milky Way mass disc galaxies. Font et
al. have recently demonstrated that these simulations are able to successfully
reproduce the satellite luminosity functions and the metallicity and surface
brightness profiles of the spheroids of the Milky Way and M31. A key to the
success of the simulations is a significant contribution to the spheroid from
stars that formed in situ. While the outer halo is dominated by accreted stars,
stars formed in the main progenitor of the galaxy dominate at r < ~30 kpc. In
the present study we show that this component was primarily formed in a
proto-disc at high redshift and was subsequently liberated from the disc by
dynamical heating associated with mass accretion. As a consequence of its
origin, the in situ component of the spheroid has different kinematics (namely
net prograde rotation with respect to the disc) than that of the spheroid
component built from the disruption of satellites. In addition, the in situ
component has a flattened distribution, that is due in part to its rotation. We
make comparisons with measurements of the shape and kinematics of local
galaxies, including the Milky Way and M31, and stacked observations of more
distant galaxies. We find that the simulated disc galaxies have spheroids of
the correct shape (oblate with a median axis ratio of ~0.6 at radii of < ~30
kpc, but note there is significant system-to-system scatter in this quantity)
and that the kinematics show evidence for two components (due to in situ vs.
accreted), as observed. Our findings therefore add considerable weight to the
importance of dissipative processes in the formation of stellar haloes and to
the notion of a 'dual stellar halo'.
