Abstract
This paper deals with the transverse free vibration of axially functionally graded (AFG) cantilever columns under the influence of axial compressive load. The columns possessing a regular polygon in their cross-section are tapered and their material properties vary along the axis of the column. An emphasis is placed on the columns with constant volume for admissible geometries and materials. The governing differential equation of the problem is derived and solved using the direct integral approach in conjunction with the determinant search technique. The obtained results are in good agreement with those in the available literature and computed by finite element analysis. Numerical examples for the natural frequency and mode shape of the columns are presented to investigate the effects of parameters related to geometrical nonuniformity and material inhomogeneity.
Highlights
In a variety of structural engineering applications, columns are often built as one of the most important main components by which axial compressive forces, one of the main types of external loading, are supported [1]
This paper presents a unique numerical approach for analyzing the free vibration of axially functionally graded (AFG) cantilever columns
For integrating differential equations to calculate the mode shape, the Runge-Kutta method [20], a direct integral method, was used, and for computing the frequency parameter Ci, the determinant search method enhanced by the Regula-Falsi method [20]
Summary
In a variety of structural engineering applications, columns are often built as one of the most important main components by which axial compressive forces, one of the main types of external loading, are supported [1]. Li [6] studied critical buckling loads of a cantilever column with a piecewise element, restrained by the elastic foundation; Huang and Li [7] researched a new approach for the free vibration of tapered beams with supported, both clamped, clamped-pinned, and cantilevered end conditions, respectively, where the Fredholm integral equations were used in the mathematical formulations; Shahba et al [8]. By using the equilibrium of free body diagram of the column element subjected to the transverse and rotatory inertia loadings, the differential equation governing the mode shape of vibrating columns is derived with its boundary conditions. The above fourth order ordinary differential Equation (34) with boundary conditions, Equations (36) and (37), governs the free vibration of AFG cantilever columns with a regular polygon cross-section and constant volume. In Equation (34), the taper type, side number k, modular ratio m, taper ratio n, volume ratio λ, and load parameter p are input parameters, while Ci is the eigenvalue which is calculated with its mode shape (ξi , ηi ), using appropriate numerical methods
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