A high-precision curvature constrained Bernoulli–Euler planar beam element for geometrically nonlinear analysis

In this work, a high-precision curvature constrained Bernoulli–Euler beam element based on the gradient deficient beam elements (GDBE) of the absolute nodal coordinate formulation (ANCF) is developed for precisely and efficiently tackling the geometrically nonlinear analysis of straight and strongly curved planar thin beam structures. Firstly, this research proposes a vector interpolation scheme named “curvature constrained interpolation method” (CCIM) which can present comprehensive and accurate curve properties including Frenet frame, position, gradient and especially, curvature at base points. Since the correctness and superiority of the CCIM are proven, it is therefore utilized as the shape function of the proposed element, which ensures the second-order accuracy. With the accurate Frenet frame provided by the CCIM, the strain energy including axial strain and bending strain of a beam element is rigorously derived by applying the definition of the Green–Lagrange strain tensor in continuum mechanics with the Bernoulli–Euler beam assumption. The sign of curvature in the bending strain term is determined by a referenced vector. This revision makes the proposed element feasible when solving the configuration which presents different orientations of concavity or the deformed configuration where the orientation of concavity reverses with comparison to the initial configuration. Due to the constant global reference coordinate, the elastic force and its Jacobian matrix are expressed in an elegant form, which can be calculated by the Gauss quadrature integration conveniently. The efficiency and high accuracy of the proposed element are validated by several nonlinear benchmark problems, which is especially reflected in considerably fewer numbers of elements and continuous, exact strain result between adjacent elements with comparison to relevant examples from literature and ABAQUS. After comprehensive demonstrations, it can be concluded that the proposed element is accurate and effective enough for modelling planar thin beam structures experiencing large deformation and large rotation.