Free vibration analysis of the cracked post-buckled axially functionally graded beam under compressive load

Document Type : Research Paper


Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran


This paper aims to discuss the vibration analysis of the post-buckled cracked axially functionally graded (AFG) beam. The nonlinear equations of motion of the Euler-Bernoulli beam are derived using the equilibrium principles. Then, these differential equations are converted into a set of algebraic ones using the differential quadrature (DQ) method and solved by an arc-length strategy. The resulted displacement field from the post-buckling analysis is assumed to be the equilibrium state of vibration analysis, and an eigenvalue problem is derived. By solving this linear eigenvalue problem, both the natural frequencies and mode shapes of the beam are calculated. The validation of results in comparison with a similar work shows a good agreement. The effect of several parameters such as the extensible and inextensible clamped-clamped boundary conditions, initial geometric imperfection, crack’s depth, and crack’s location on the natural frequencies and mode shapes are investigated in detail.


Volume 52, Issue 2
June 2021
Pages 256-270
  • Receive Date: 04 March 2021
  • Revise Date: 06 May 2021
  • Accept Date: 06 May 2021