Equivalently analytical solution for the large deformation of slender beams under follower loads: a second-order ANCF approach

Follower loads interact with the object's configuration, and the two are unknown variables in geometrically nonlinear analysis. This work aims to solve the large deformation of slender beams under follower loads via the Bernoulli–Euler beam element of absolute nodal coordinate formulation. The second-order approximate function of the beam centerline is utilized to model the beam configuration and the Frenet frame. The elastic force vector and its tangent stiffness matrix are formulated following the decoupling elastic line approach. The constraint of adjacent elements is imposed using Lagrange multipliers. The exact kinematics characteristics of typical follower loads are demonstrated; sequentially, the external force vectors and the load stiffness matrices are rigorously derived based on the configuration and the Frenet frame modelled by the second-order approximate function. Benchmark problems of straight and curved beams under different follower loads are solved, wherein the accuracy and efficiency of the proposed approach are validated as compared to relevant analytical and numerical solutions. It is found that the proposed element yields equivalently analytical nodal solutions, including position, rotation and curvature; considerably fewer proposed elements are used as compared with other finite element methods; the load stiffness matrix of the bending moment raises computational efficiency of the proposed element; the obtained solutions are more accurate than that of the B22 element of ABAQUS when solving pressure loading cases. By further extending the proposed approach, it can enable the unveiling of the spatio-temporal evolution characteristic of slender structures and follower loads from the equivalently analytical perspective.