  Gravity-induced non-Gaussianity in the large-scale structure of the Universe,
characterised by higher-order statistics such as the bispectrum (three-point
cumulant), is expected to contain rich cosmological information. A measurement
of the bispectrum will not only improve the cosmological constraints, but also
give us the possibility to probe gravity on cosmological scales. In this paper,
we present a framework to numerically calculate the one-loop matter bispectrum
based on standard perturbation theory (SPT). This approach allows general
modifications to the standard $\Lambda$CDM model to be easily implemented. We
demonstrate the performance of the bispectrum calculation in three
representative cases, namely the Vainshtein-screened Dvali-Gabadadze-Porrati
(DGP) model, the chameleon-screened Hu-Sawicki $f(R)$ model and the
phenomenological dark scattering (DS) momentum-exchange model. The predicted
bispectra are then compared with measured results from a set of cosmological
$N$-body simulations, and the impact of possible systematics arising from
simplified or approximate treatments in the perturbative calculation is studied
in detail. We find that the one-loop bispectrum calculation offers
significantly more information on general screening and momentum exchange
effects than the leading-order bispectrum calculation. Further, the accuracy of
the one-loop prediction is shown to be comparable to non-linear fitting
formulas over a wide range of wavenumbers ($k\lesssim0.3\,h$$\mbox{Mpc}^{-1}$)
even at lower redshifts, $z\lesssim 1$.
