Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams
TL;DR: In this paper, a novel method based on mechanical/structural principles is introduced for free vibration analysis of arbitrarily tapered Timoshenko beams in preference to primarily mathematically based methodologies.
About: This article is published in Finite Elements in Analysis and Design.The article was published on 2010-10-01. It has received 60 citations till now. The article focuses on the topics: Timoshenko beam theory & Finite element method.
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TL;DR: In this article, the free vibration and stability analysis of axially functionally graded tapered Timoshenko beams is studied through a finite element approach, where exact shape functions for uniform homogeneous Timoshenko beam elements are used to formulate the proposed element.
Abstract: Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams are studied through a finite element approach. The exact shape functions for uniform homogeneous Timoshenko beam elements are used to formulate the proposed element. The accuracy of the present element is considerably improved by considering the exact variations of cross-sectional profile and mechanical properties in the evaluation of the structural matrices. Carrying out several numerical examples, the convergence of the method is verified and the effects of taper ratio, elastic constraint, attached mass and the material non-homogeneity on the natural frequencies and critical buckling load are investigated.
253 citations
Additional excerpts
...The increase of the fundamental frequency for a homogeneous cantilever beam has been previously reported for both Euler–Bernoulli [17,18] and Timoshenko [25] beams....
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...Thus, numerical methods have been used among which finite element method has gained a more prominent position [22–25]....
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TL;DR: In this paper, a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section is presented. And the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-root.
Abstract: This paper presents a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation. Moreover, all physical quantities can be expressed in terms of the solution of the resulting equation. Making use of power series for unknown function, we can transform the single equation to a system of linear algebraic equations and will get a characteristic equation in natural frequencies for different boundary conditions. An advantage of the suggested approach is that the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-roots. Several examples of estimating natural frequencies for axially grade beams and non-uniform beams are presented, which show that our method has fast convergence and obtained numerical results have high accuracy.
160 citations
Cites background from "Basic displacement functions for fr..."
...Recently, Attarnejad and co-workers studied the free vibration of arbitrarily tapered Timoshenko beams with classical and non-classical boundary conditions based on basic displacement functions [18] and finite element approach [19], respectively....
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TL;DR: In this article, the free vibration of non-uniform functionally graded beams is analyzed via the Timoshenko beam theory, where bending stiffness and distributed mass density are assumed to obey a unified exponential law.
101 citations
TL;DR: In this article, a co-rotational beam element taking the effects of the material inhomogeneity, shear deformation and nonuniform cross section into account is formulated and employed in computing the response of the beams.
Abstract: The large displacement response of tapered cantilever beams made of axially functionally graded material is investigated by the finite element method. A co-rotational beam element taking the effects of the material inhomogeneity, shear deformation and nonuniform cross section into account is formulated and employed in computing the response of the beams. The numerical results show that the large displacement response of the beam is governed by the material distribution, the taper ratio and taper type. The axial displacement at the free end of the beam is most sensitive to the taper ratio, and the transverse displacement at the point is least affected by this parameter. The influence of the length to height ratio is also investigated and highlighted.
69 citations
TL;DR: In this article, a simple beam model based on standard Timoshenko kinematics is derived, describing accurately the effects of non-prismatic geometry on the beam behavior and motivating equation's terms with both physical and mathematical arguments.
63 citations
References
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Book•
01 Jan 2002
TL;DR: In this paper, the authors present a review of the work, energy, and variational calculus of solid mechanics and their application in the analysis of plate models. But their focus is on the theory and analysis of plates.
Abstract: Preface xv 1 Introduction 1 2 Mathematical Preliminaries 8 3 Review Of Equations Of Solid Mechanics 48 4 Work, Energy, And Variational Calculus 79 5 Energy Principles Of Structural 133 6 Dynamical Systems: Hamilton's Principle 177 7 Direct Variational Methods 204 8 Theory And Analysis Of Plates 299 9 The Finite Element Method 433 10 Mixed Variational Formulations 502 Answers / Solutions to Selected Problems 544 Index 583 About the Author 591
926 citations
TL;DR: In this article, Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of a beam.
Abstract: Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where A, GJ and I have their usual meanings; y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of the beam. Numerical checks give better than seven-figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.
154 citations
TL;DR: In this paper, the in-plane and out-of-plane free vibrations of a rotating Timoshenko beam are analyzed by means of a finite element technique, where the beam is discretized into a number of simple elements with four degrees of freedom each.
123 citations
TL;DR: In this paper, the authors developed an exact dynamic stiffness matrix of a composite beam with the effects of axial force, shear deformation and rotatory inertia taken into account, for an axially loaded composite Timoshenko beam.
122 citations