The paper studies a bundle of oriented carbon nanotubes (CNTs) under the transverse loading under the plane deformation conditions within the framework of a molecular dynamics model with a reduced number of degrees of freedom. The model takes into account CNT wall stretching and bending, as well as van der Waals interactions. Each CNT is represented by a ring of atoms with two degrees of freedom in the plane of the ring. The discrete nature of the model allows describing the large curvature of the CNT wall and the destruction of CNTs at very high pressure. CNT crystal equilibrium structures are obtained under the strain-controlled biaxial loading. Separate CNTs of a sufficiently large diameter have two equilibrium states: with a round and collapsed cross section. Small-diameter CNTs in the free state can only have a circular cross section. The study identified the presence of two phase transitions observed during biaxial compression of a CNT bundle. The first transformation similar to phase transition of the second order leads to ellipticization of CNT cross sections. As a result of the second transition of the first order, bundled CNTs appear in the beam, the proportion of which gradually increases with the increase in compressive strain. The authors calculated beam elasticity constants such as Young’s moduli, shear modulus, and Poisson’s ratios. The study shows that one of the equilibrium structures (with elliptical CNT cross sections) has the property of a partial auxetic, that is, it has a negative Poisson’s ratio under uniaxial loading in a certain direction. The proposed chain model can be effectively applied to analyze physical and mechanical properties of bundles of single-walled or multi-walled CNTs under the plane deformation conditions, and after simple modifications, it can be used to similar structures made of other two-dimensional nanomaterials.