  Short (inner) bars of sub-kiloparsec radius have been hypothesized to be an
important mechanism for driving gas inflows to small scales, thus feeding
central black holes. Recent numerical simulations have shown that the growth of
central black holes in galaxies can destroy short bars, when the black hole
reaches a mass of $\sim 0.1\%$ of the total stellar mass of the galaxy. We
study $N$-body simulations of galaxies with single and double bars to track the
long-term evolution of the central stellar mass distribution. We find that the
destruction of the short bar contributes significantly to the growth of the
bulge. The final bulge mass is roughly equal to the sum of the masses of the
initial pseudo bulge and short bar. The initially boxy/peanut-shaped bulge of
S\'ersic index $n\lesssim1$ is transformed into a more massive, compact
structure that bears many similarities to a classical bulge, in terms of its
morphology ($n \approx 2$), kinematics (dispersion-dominated, isotropic), and
location on standard scaling relations (Kormendy relation, mass-size relation,
and correlations between black hole mass and bulge stellar mass and velocity
dispersion). Our proposed channel for forming classical bulges relies solely on
the destruction of short bars without any reliance on mergers. We suggest that
some of the less massive, less compact classical bulges were formed in this
manner.
