Analytical Solutions for Timoshenko Beam-Columns on Elastic Foundations

The study of the dynamic behaviour of non-uniform thickness circular plate resting on elastic foundations is very imperative in designing structural systems. This present research investigates the free vibration analysis of varying density and non-uniform thickness isotropic circular plates resting on Winkler and Pasternak foundations. The governing differential equation is analysed using the Galerkin method of weighted residuals. Linear and nonlinear case is considered, the surface radial and circumferential stresses are also determined. Thereafter, the accuracy and consistency of the analytical solutions obtained are ascertained by comparing the existing results available in pieces of literature and confirmed to be in a good harmony. Also, it is observed that very accurate results can be obtained with few computations. Issues relating to the singularity of circular plate governing equations are handled with ease. The analytical solutions obtained are used to determine the influence of elastic foundations, homogeneity and thickness variation on the dynamic behaviour of the circular plate, the effect of vibration on a free surface of the foundation as well as the influence of radial and circumferential stress on mode shapes of the circular plate considered. From the results, it is observed that a maximum of 8.1% percentage difference is obtained with the solutions obtained from other analytical methods. Furthermore, increasing the elastic foundation parameter increases the natural frequency. Extrema modal displacement occurs due to radial and circumferential stress. Natural frequency increases as the thickness of the circular plate increases, Conversely, a decrease in natural frequency is observed as the density varies. It is envisioned that; the present study will contribute to the existing knowledge of the classical theory of vibration.

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Recursive Differentiation Method for Boundary Value Problems: Application to Analysis of a Beam-Column on an Elastic Foundation

Journal of Theoretical and Applied Mechanics ◽

10.2478/jtam-2014-0010 ◽

2014 ◽

Vol 44 (2) ◽

pp. 57-70 ◽

Cited By ~ 4

Author(s):

Mohamed Taha

Keyword(s):

Differential Equations ◽

Analytical Solutions ◽

Boundary Value ◽

Adomian Decomposition Method ◽

Buckling Mode ◽

Beam Columns ◽

Differentiation Method ◽

Adomian Decomposition ◽

Different Types ◽

Finite Domains

Abstract In the present work, the recursive differentiation method (RDM) is introduced and implemented to obtain analytical solutions for differential equations governing different types of boundary value prob- lems (BVP). Then, the method is applied to investigate the static behaviour of a beam-column resting on a two parameter foundation subjected to different types of lateral loading. The analytical solutions obtained using RDM and Adomian decomposition method (ADM) are found similar but the RDM requires less mathematical effort. It is indicated that the RDM is reliable, straightforward and efficient for investigation of BVP in finite domains. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. However, it is indicated that, in the case of beam-columns resting on foundations, the beam-column may be buckled in a higher mode rather than a lower one, then the critical load in that case is that accompanies the higher mode. This result is very important to avoid static instability as it is widely common that the buckling load of the first buckling mode is always the smaller one, which is true only in the case of the beam-columns without foundations.

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Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass

International Journal of Mechanical Sciences ◽

10.1016/j.ijmecsci.2019.01.033 ◽

2019 ◽

Vol 153-154 ◽

pp. 21-35 ◽

Cited By ~ 12

Author(s):

Ismail Esen

Keyword(s):

Dynamic Response ◽

Timoshenko Beam ◽

Functionally Graded ◽

Moving Mass ◽

Variable Velocity ◽

Elastic Foundations ◽

Two Parameter

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A numerical method for static and free-vibration analysis of non-uniform Timoshenko beam–columns

Canadian Journal of Civil Engineering ◽

10.1139/l05-109 ◽

2006 ◽

Vol 33 (3) ◽

pp. 278-293 ◽

Cited By ~ 3

Author(s):

Z Canan Girgin ◽

Konuralp Girgin

Keyword(s):

Free Vibration ◽

Numerical Method ◽

Timoshenko Beam ◽

Dynamic Stiffness ◽

Geometrical Nonlinearity ◽

Free Vibration Analysis ◽

Winkler Foundation ◽

Beam Columns ◽

Geometrical Properties ◽

Stiffness Matrices

A generalized numerical method is proposed to derive the static and dynamic stiffness matrices and to handle the nodal load vector for static analysis of non-uniform Timoshenko beam–columns under several effects. This method presents a unified approach based on effective utilization of the Mohr method and focuses on the following arbitrarily variable characteristics: geometrical properties, bending and shear deformations, transverse and rotatory inertia of mass, distributed and (or) concentrated axial and (or) transverse loads, and Winkler foundation modulus and shear foundation modulus. A successive iterative algorithm is developed to comprise all these characteristics systematically. The algorithm enables a non-uniform Timoshenko beam–column to be regarded as a substructure. This provides an important advantage to incorporate all the variable characteristics based on the substructure. The buckling load and fundamental natural frequency of a substructure subjected to the cited effects are also assessed. Numerical examples confirm the efficiency of the numerical method.Key words: non-uniform, Timoshenko, substructure, elastic foundation, geometrical nonlinearity, stiffness, stability, free vibration.

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