Journal of Structural and Construction Engineering

Journal of Structural and Construction Engineering

Development of the exact stiffness matrix for a curved beam element with quadratic parabolic geometry: An energy-based approach

Document Type : Original Article

Authors
Ali Keshmiri 1
Amirreza Masoodi 2
Mohammad Rezaiee-Pajand 3
1 M.Sc. student, Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
2 Assistant Professor, Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
3 Professor, Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
10.22065/jsce.2025.530704.3762
Abstract
Accurate analysis of curved structures, especially in engineering applications such as bridges, curved frames, and space structures, requires high-precision mechanical models. In this study, for the first time, an explicit method is presented for expanding the stiffness matrix of a curved beam element with a quadratic parabolic geometry. This method is developed based on energy principles and the application of the superposition principle. Since strain energy is a function of internal forces, its computation involves integration along the element. Additionally, shear deformations are considered in the formulation of the stiffness matrix components. Due to the complexity of the quadratic parabolic geometry, exact evaluation of the required integrals poses significant computational challenges. To overcome this and enhance both the accuracy and efficiency of the formulation, Gaussian numerical integration is employed, providing a precise approximation of the required quantities. Consequently, the stiffness matrix of the parabolic beam element with six degrees of freedom is explicitly obtained. This matrix is directly applicable to bending analysis and free vibration of curved structures. Finally, to validate the formulation and demonstrate the accuracy and efficiency of the proposed parabolic curved element, various numerical examples are solved. These examples provide the displacements, rotations, and natural frequencies of parabolic curved beams under different geometries, loading conditions, and boundary configurations.
Keywords
Explicit stiffness matrix
Parabolic curved beam
Natural frequency
Bending behavior
Energy approach
Gaussian integration

Subjects

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  • Receive Date 24 June 2025
  • Revise Date 02 October 2025
  • Accept Date 04 November 2025