Geometric nonlinear cubic spline beam element with lateral and axial deformation effects

A cubic spline Bernoulli-Euler beam element was proposed to accurately analyze the large displacement, large rotation and small strain problems based on the updated Lagrangian formula and the moving coordinate method. The deformation field was built using the cubic spline function, its curvature DoF was eliminated by static condensation method, the geometric nonlinear tangent stiffness matrix of the beam element was derived with consideration of the second-order effects. The nonlinearity of the axial length change caused by bending deformation was considered, the additional stiffness caused by the bowing effects was derived, and the tangent stiffness matrix of the beam element was amended. The moving coordinate system of the hinged-hinged beam element was built on the updated configuration after deformation, and the large displacement total equilibrium equations of the cubic spline beam element were derived. The correctness and efficiency of the proposed element were shown by several typical numerical examples.