We explore the effects of background cosmology on large-scale structures with non-spherical symmetry by using the concept of quasi-equilibrium which allows certain internal properties (e.g. angular velocity) of the bodies to change with time. In accordance with the discovery of the accelerated phase of the universe, we model the cosmological background by two representative models: the ΛCDM model and the Chaplygin gas model. We compare the effects of the two models on various properties of large astrophysical objects. Different equations of state are also invoked in the investigation.