Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

Abstract

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

“…The model accommodates (a) the role of size-effect through the MCST, (b) the lateral inertial effect in the axial vibration via the Rayleigh–Love (RL) theory and (c) the bending moment-induced/shear deformation effects in the transverse vibration. The combination of the RL theory with the shear deformable beam theory is motivated by two recent studies (Banerjee and Ananthapuvirajah, 2018, 2019).…”

Coverage intensity of optimal sensors for common, isolated, and integrated steel structures using novel approach of FEM-MAC-TTFD

“…The stiffness matrix is partitioned to separate the actions associated with two ends of the member. 33 In this study, lead rubber bearing (LRB) system was selected as a BI. 34 The geometry of an LRB is shown in Figure 1.…”

“…The stiffness matrix is partitioned to separate the actions associated with two ends of the member. 33…”

Dynamic Stiffness Formulation for Free Vibration of Truncated Conical Shell and Its Combinations with Uniform Boundary Restraints

“…In recent years, an elegant modeling method called the dynamic stiffness method (DSM) [25] has been well developed and widely used in vibration analysis of beams, plates and shells structures [26][27][28][29][30][31][32][33][34][35][36]. In the DSM, the frequency-dependent shape functions are derived directly from the governing equation.…”

“…However, not all kind of structures can obtain exact shape function from the governing equation. us, the DSM is mostly used in one-dimensional structure like beams [26,27] and Levy-type plate [37]. Until the last few years, it is reported that the DSM has been used to establish the dynamic model of rectangular plate with general boundary conditions [38,39].…”