This　paper　presents　the　formulation　of　the　equilibrium　problem　of　a　three-bar　truss　in　the　nonlinear　context　of　finite　elasticity.　The　bars　are　composed　of　a　homogeneous,　isotropic,　and　compressible　hyperelastic　material.　The　equilibrium　equations　in　the　deformed　configuration　are　derived　under　the　assumption　of　homogeneous　deformations　and　the　stability　of　the　solutions　is　assessed　through　the　energy　criterion.　The　general　formulation　is　then　specialized　for　a　compressible　Mooney–Rivlin　material.　The　results　for　both　vertical　and　horizontal　load　cases　show　unexpected　post-critical　behaviors　involving　several　branches,　stable　asymmetrical　configurations,　bifurcation,　and　snap-through.　The　three-bar　truss　studied　here　is　not　only　a　benchmark　test　for　the　numerical　analysis　of　nonlinear　truss　structures,　but　also　a　representative　system　for　the　unit　cell　of　the　graphene　hexagonal　lattice.　Therefore,　an　application　to　graphene　is　performed　by　simulating　the　covalent　bonds　between　carbon　atoms　as　the　bars　of　the　truss,　characterized　by　the　modified　Morse　potential.　The　results　provide　insights　on　the　internal　mechanisms　that　take　place　when　graphene　undergoes　large　in-plane　deformations,　whose　influence　should　be　considered　when　developing　molecular　mechanics　and　continuum　models　in　nonlinear　elasticity.　©　The　Author(s)　2019.