  We present a study of the effective (half-light) radii and other structural
properties of a systematically selected sample of young, massive star clusters
(YMCs, $\geq$$5\times10^3$ M$_{\odot}$ and $\leq$200 Myr) in two nearby spiral
galaxies, NGC 628 and NGC 1313. We use Hubble Space Telescope WFC3/UVIS and
archival ACS/WFC data obtained by the Legacy Extragalactic UV Survey (LEGUS),
an HST Treasury Program. We measure effective radii with GALFIT, a
two-dimensional image-fitting package, and with a new technique to estimate
effective radii from the concentration index (CI) of observed clusters. The
distribution of effective radii from both techniques spans $\sim$0.5-10 pc and
peaks at 2-3 pc for both galaxies. We find slight positive correlations between
effective radius and cluster age in both galaxies, but no significant
relationship between effective radius and galactocentric distance. Clusters in
NGC 1313 display a mild increase in effective radius with cluster mass, but the
trend disappears when the sample is divided into age bins. We show that the
vast majority of the clusters in both galaxies are much older than their
dynamical times, suggesting they are gravitationally bound objects. We find
that about half of the clusters in NGC 628 are underfilling their Roche lobes,
based on their Jacobi radii. Our results suggest that the young, massive
clusters in NGC 628 and NGC 1313 are expanding due to stellar mass loss or
two-body relaxation and are not significantly influenced by the tidal fields of
their host galaxies.
