This work presents an Equilibrium Finite Element Method (EFEM) previously used in structural Mechanics [1] to solve the Steady--state Stokes problem with the Cauchy--Stress tensor as unknown and fluid--structure problem (stationary case). The Stokes system is rewritten as the stress system and recast as a constrained minimization problem : \begin{equation} \label{eq1} \begin{aligned} \min_{\boldsymbol{\sigma}} \; J(\boldsymbol{\sigma}) &= \frac{1}{4\eta} \int_{\Omega} \boldsymbol{\sigma}:\boldsymbol{\sigma}\,\mathrm{d}\Omega - \frac{1}{6\eta} \int_{\Omega} \operatorname{tr}(\boldsymbol{\sigma})^2\,\mathrm{d}\Omega - \int_{\partial \Omega_u} \boldsymbol{\sigma}\mathbf{n}\cdot\overline{\mathbf{v}}\,\mathrm{d}S \\ \text{s.t.} \quad & \operatorname{div}\boldsymbol{\sigma} = \mathbf{0} \quad \text{in } \Omega, \\ & [|\boldsymbol{\sigma}|]\mathbf{n} = \mathbf{0} \quad \text{on } \Omega_{\Delta}, \\ & \boldsymbol{\sigma}\mathbf{n} = \mathbf{t} \quad \text{on } \partial \Omega_t . \end{aligned} \end{equation} In each element a linear interpolation of the stress tensor is adopted, while the velocity field $\mathbf{v}$ appears as the Lagrange multiplier associated with the constraint $[|\boldsymbol{\sigma}|]\mathbf{n} = \mathbf{0}$. For the two-dimensional channel flow around a cylinder case, the results are compared to those obtained using the \OpenFOAM solver. A convergence analysis was conducted over a restricted portion of the computational domain, specifically upstream of the cylinder, where the flow remains relatively smooth and the error captures the intrinsic accuracy of the discretization showing a quadratic and linear convergence for the pressure variable using EFEM and first-order Finite Volume method, respectively. Finally, the equilibrium formulation is extended to static fluid-structure interaction problem, by combining the energy expressions derived from fluid and structure models, while ensuring the continuity of the stress field at the fluid-structure interface.