A hybrid finite element approach for linear and nonlinear dynamic beam equations: Implementation and validation

This study presents a novel hybrid numerical method based on Rothe’s method for solving unsteady damped Euler–Bernoulli beam equations. Initially, the problem is transformed into a system of coupled first-order initial value problems and an appropriate finite difference approach is employed to discretize the time derivative, which eventually reduces the problem to a sequence of steady-state systems. The resulting family of steady-state problems is then solved iteratively using a finite element method with Hermite cubic basis functions. Through this procedure, approximations of the beam’s displacement, slope, transverse velocity and rate of slope change are simultaneously obtained. A comparison with the exact solutions is conducted to evaluate the accuracy of the proposed approach. Plots in logarithmic scale are used to examine the influence of discretization parameters on both types of problems: steady and unsteady. Moreover, adaptability and robustness of the method to non-linear equations are demonstrated through a benchmark model problem. Finally, a solvability analysis for the weak solution of the steady problem is presented in Appendix, along with a detailed error analysis that supports the numerical results in L2 norm.