Exact natural frequencies and buckling load of functionally graded material tapered beam-columns considering semi-rigid connections

Abstract

An exact free vibration and buckling analysis of a tapered beam-column with general connections is calculated. All structural elements are made of functionally graded material. In this study, a power function is assumed for the variation of the elastic modulus along the cross-section’s height. Extensional-coupling effects are considered in the proposed formulation. By solving the related fourth-order differential equation, the exact answers are obtained. The Bessel functions are utilized in this solution. In addition, the effect of connection flexibility is assessed by considering rotational and transitional springs at the two ends of the element. The suggested stiffness matrix can analyze the free vibration and buckling of a frame made of functionally graded material tapered beam-columns with semi-rigid connections and supports. To validate the proposed formulation, several benchmarks and novel examples are separately analyzed. All numerical findings clearly demonstrate the usefulness of the authors’ scheme.

Free Vibration Analysis of Axially Functionally Graded (AFG) Cantilever Columns of a Regular Polygon Cross-Section with Constant Volume

“…There are various kinds of functions for Young's modulus E in the literature: linear [2,3,9,12,14,17], trigonometric [4], polynomial [5,[14][15][16], piecewise [6], exponential [7,10], and periodic [14] functions, etc. The linear function is selected in this study, and then the function of E at the coordinate x is expressed as [17]…”

“…For the variable function of the mass density ρ, it is usual that ρ is equal to E [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and then the density ratio is the same of modular ratio m (see Figure 1c). Therefore, the mass density ratio m defined as a ratio of ρ f to ρ c and the linear function of ρ at the coordinate x can be written as…”

“…Using Equations (16) and (23), the variable function of flexural rigidity R f (= EI) at the coordinate x is established:…”

“…The following references and their citations include the mathematical models and historical reviews of this subject. The typical studies on free vibration of AFG columns are briefly reviewed here in chronological order: Akgoz and Civalek [3] studied natural frequencies of a linearly tapered microbeam based on the modified couple stress theory; Calio and Elishakoff [4] investigated the closed-form solutions of natural frequency for a simply supported beam-column with an elastically guided end condition at one end, where the trigonometric functions of the Young's modulus in the mathematical formulation were considered; Li [5] studied the static and dynamic behaviors of a prismatic beam, including the rotatory inertia and shear deformation; Singh and 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 simply 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] investigated the free vibration and stability of a non-prismatic column with classical and non-classical boundary conditions; Shahba and Rajasekaran [9] studied the free vibration and stability of tapered beams, in which governing equations for the free vibration and buckling were solved by the differential quadrature element method (DQEM); Kukla and Rychlewska [10] dealt with beams with both ends fixed, which were fabricated from two different FGMs for analyzing free vibration; Yilmaz et al [11] investigated the buckling loads of non-prismatic columns restrained by the elastic foundation using the differential quadrature method (DQM); Chandran and Rajendran [12] studied closed-form solutions of the buckling load of a prismatic cantilever column using the principle of conservation energy; Shafiei et al [13] studied non-linear vibrations of a linearly tapered microbeam with a square cross-section, where the governing equations were solved by the differential quadrature method (DQM); Ranganathan et al [14] studied the buckling of slender columns that determined the buckling loads by the linear perturbation method together with the Rayleigh-Ritz method and investigated the maximum buckling loads under the same average mean Young's modulus; Elishakoff et al [15] studied the buckling and vibrations of a column sharing Duncan's mode shapes and assuming a fifth order polynomial; Rezaiee and Masoodi [16] investigated closed-form solutions of the natural frequencies and buckling loads of tapered beam-columns supported by semi-rigid connections; and Lee and Lee [17] studied the free vibration and buckling of tapered cantilever columns with square and circular cross-sections. As discussed above, FG materials developed in 1984 are of particular interest in dealing with the free vibration of FG columns.…”

A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation

“…Akgoz and Civalek [53] investigated the buckling behavior of tapered microbeams by means of strain gradient theories, and applied the Rayleigh-Ritz method to solve the problem in terms of buckling load for different non-uniformity ratios. Other applications of the strain gradient theory for the vibration and/or buckling analysis of small-scale beams with a non-uniform geometry and material, can be found in References [54][55][56][57][58][59], where, in most cases, the differential quadrature method has been applied to solve the governing equations of the problem. In the further work by , the authors analyzed the flapwise bending vibration response of a tapered rotating nanocantilever beam through the Eringen's nonlocal elastic theory, while using the pseudospectral collocation method based on Chebyshev polynomials to solve the problem.…”