  The oscillator bases expansion stands as an efficient approximation method
for the time-independent Schr\"odinger equation. The method, originally
formulated with one non-linear variational parameter, can be extended to
incorporate two such parameters. It handles both non- and semi-relativistic
kinematics with generic two-body interactions. In the current work, focusing on
systems of three identical bodies, the method is generalised to include the
management of a given class of three-body forces. The computational cost of
this generalisation proves to not exceed the one for two-body interactions. The
accuracy of the generalisation is assessed by comparing with results from
Lagrange mesh method and hyperspherical harmonic expansions. Extensions for
systems of $N$ identical bodies and for systems of two identical particles and
one distinct are also discussed.
