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Background <p>A new beam p-finite element formulation with three degrees of freedom per node was developed and used to numerically investigate the dynamic response of Euler–Bernoulli beams under moving loads. The curvature is introduced as an additional degree of freedom alongside displacement and rotation. Since curvature is directly related to the bending moment and the corresponding strain energy, its inclusion provides a more consistent representation of the bending behavior, ensures C2 continuity within the element domain, and improves both the accuracy and convergence of beam vibration analyses.A new beam p-finite element formulation with three degrees of freedom per node was developed and used to numerically investigate the dynamic response of Euler–Bernoulli beams under moving loads. The curvature is introduced as an additional degree of freedom alongside displacement and rotation. Since curvature is directly related to the bending moment and the corresponding strain energy, its inclusion provides a more consistent representation of the bending behavior, ensures C2 continuity within the element domain, and improves both the accuracy and convergence of beam vibration analyses.</p> Method <p>The governing equation of the beams under moving loads is developed. The variable fields are represented through out-of-plane shape functions formulated from the standard finite element basis, augmented by a variable number of additional shape functions to describe the internal degrees of freedom of flexible beams. In this article, the precise time integration method (PTIM), in conjunction with the p-finite element formulation, is proposed for calculating the response of beams subjected to moving loads.</p> Results and conclusion <p>The convergence properties of the beam p-element under moving loads and boundary conditions on the dynamic response are studied. It is shown that using this element the order of the resulting matrices in the FEM is considerably reduced leading to a significant decrease in computational effort.</p>

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High Order p Finite Element with Curvature DOF for Dynamic Analysis of Beams under Moving Loads

  • 期刊论文

  • 发表日期：

    2026年5月4日

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  • Yazid Barka,
  • Sidi Mohammed Hamza Cherif

摘要

Background

A new beam p-finite element formulation with three degrees of freedom per node was developed and used to numerically investigate the dynamic response of Euler–Bernoulli beams under moving loads. The curvature is introduced as an additional degree of freedom alongside displacement and rotation. Since curvature is directly related to the bending moment and the corresponding strain energy, its inclusion provides a more consistent representation of the bending behavior, ensures C2 continuity within the element domain, and improves both the accuracy and convergence of beam vibration analyses.A new beam p-finite element formulation with three degrees of freedom per node was developed and used to numerically investigate the dynamic response of Euler–Bernoulli beams under moving loads. The curvature is introduced as an additional degree of freedom alongside displacement and rotation. Since curvature is directly related to the bending moment and the corresponding strain energy, its inclusion provides a more consistent representation of the bending behavior, ensures C2 continuity within the element domain, and improves both the accuracy and convergence of beam vibration analyses.

Method

The governing equation of the beams under moving loads is developed. The variable fields are represented through out-of-plane shape functions formulated from the standard finite element basis, augmented by a variable number of additional shape functions to describe the internal degrees of freedom of flexible beams. In this article, the precise time integration method (PTIM), in conjunction with the p-finite element formulation, is proposed for calculating the response of beams subjected to moving loads.

Results and conclusion

The convergence properties of the beam p-element under moving loads and boundary conditions on the dynamic response are studied. It is shown that using this element the order of the resulting matrices in the FEM is considerably reduced leading to a significant decrease in computational effort.