Motivated by modeling directional drilling dynamics where planar curved beams undergo small displacements, withstand high compression forces, and are in contact with an external wall, this paper presents an finite element method (FEM) modeling framework to describe planar curved beam dynamics under loading. The shape functions of the planar curved beam are obtained using the assumed strain field method. Based on the shape functions, the stiffness and mass matrices of a planar curved beam element are derived using the Euler–Lagrange equations, and the nonlinearities of the beam strain are modeled through a geometric stiffness matrix. The contact effects between curved beams and the external wall are also modeled, and corresponding numerical methods are discussed. Simulations are carried out using the developed element to analyze the dynamics and statics of planar curved structures under small displacements. The numerical simulation converges to the analytical solution as the number of elements increases. Modeling using curved beam elements achieves higher accuracy in both static and dynamic analyses compared to the approximation made by using straight beam elements. To show the utility of the developed FEM framework, the post-buckling condition of a directional drill string is analyzed. The drill pipe undergoes spiral buckling under high compression forces, which agrees with experiments and field observations.

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