University of Tehran Journal of Computational Applied Mechanics 2423-6713 52 2 2021 06 01 Free vibration analysis of the cracked post-buckled axially functionally graded beam under compressive load 256 270 82064 10.22059/jcamech.2021.320044.602 EN Emadaldin Sh Khoram-Nejad Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran 0000-0002-4304-6661 Shapour Moradi Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran Mohammad Shishesaz Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran 0000-0002-1892-1946 Journal Article 2021 03 04 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. https://jcamech.ut.ac.ir/article_82064_3049a04498902e0d02077c390334c29a.pdf