Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

Abstract

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy-Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy-Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the highorder theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy-Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.

Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories

“…Since the FBT violates the zero shear stress conditions on the top and bottom surfaces of the beam, a shear correction factor is required to account for the discrepancy between the actual stress state and the assumed constant stress state [4][5][6][7]. To avoid the use of a shear correction factor and have a better prediction of response of FG beams, higher-order shear deformation theories have been proposed, notable among them are the third-order theory of Reddy [8][9][10][11][12], the sinusoidal theory of Touratier [13], the hyperbolic theory of Soldatos [14], the exponential theory of Karama et al [15], and the unified formulation of Carrera [16][17]. Higher-order shear deformation theories can be developed based on the assumption of a higher-order variation of axial displacement through the depth of the beam [18][19][20][21][22][23][24] or both axial and transverse displacements through the depth of the beam (i.e.…”

Contribution to the free vibration problem of a free‐free beam with large end masses

“…Lastly, in the further authors' work the investigations conducted in this paper will be extended to the cases of Timoshenko and Reddy‐Bickford beams (see e.g. []).…”

Rotary Inertia and Higher Modes Effect on the Dynamic Response of Timoshenko Beams on Two-Parameter Elastic Foundation

“… https://dx.doi.org/10.18400/tekderg.408772 Timoshenko beams resting on Winkler type foundation and verified by the analytical results. Yeşilce et al used the DTM for the free vibration analysis of axially loaded Reddy-Bickford beams [7][8]. Sapountzakis and Kampitsis [9] investigated the nonlinear behavior of the beams partially supported by tensionless Winkler foundation.…”