A key challenge in flexible multibody dynamics is the presence of stiff differential–algebraic equations due to kinematic constraints. The Reissner–Simo and Hodges models are two equivalent representations of the dynamics of finite strain beams: the former comes typically as a displacement-based formulation, whereas the latter is intrinsic, meaning that displacement and rotations are not explicitly required. At the numerical level this equivalence is generally lost and each representation offers distinct advantages. In particular, the intrinsic formulation employs linear differential operators, which allow kinematic and dynamic boundary conditions to be imposed naturally through mixed finite elements. In this work, we present a structure-preserving discretization of the intrinsic formulation for assembling multibody systems without algebraic constraints. Through a simple numerical example of closed-loop kinematic chain, we demonstrate that the presented formulation allow to assemble systems without Lagrange multipliers, while energy and momentum are preserved using the implicit midpoint scheme.