This work presents a novel formulation and numerical strategy for the simulation of geometrically nonlinear structures. First, a non-canonical Hamiltonian (Poisson) formulation is introduced by including the dynamics of the stress tensor. This framework is developed for von-K\'arm\'an nonlinearities in beams and plates, as well as finite strain elasticity with Saint-Venant material behavior. In the case of plates, both negligible and non-negligible membrane inertia are considered. For the former case the two-dimensional elasticity complex is leveraged to express the dynamics in terms of the Airy stress function. The finite element discretization employs a mixed approach, combining a conforming approximation for displacement and velocity fields with a discontinuous stress tensor representation. A staggered, linear implicit time integration scheme is proposed, establishing connections with existing explicit-implicit energy-preserving methods. The stress degrees of freedom are statically condensed, reducing the computational complexity to solving a system with a positive definite matrix. The methodology is validated through numerical experiments on the Duffing oscillator, a von-K\'arm\'an beam, and a column undergoing finite strain elasticity. Comparisons with fully implicit energy-preserving method and the explicit Newmark scheme demonstrate that the proposed approach achieves superior accuracy while maintaining energy stability. Additionally, it enables larger time steps compared to explicit schemes and exhibits computational efficiency comparable to the leapfrog method.

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