Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
S. P. Pavlov ◽  
M. V. Zhigalov ◽  
V. A. Krysko
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Stability Loss ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Timoshenko Beams ◽  
Size Dependent ◽  
Rigid Layer ◽  
Graded Material

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

scholarly journals Vibration of deploying rectangular cross-sectional beam made of functionally graded materials

2018 ◽  
Vol 37 (4) ◽  
pp. 748-761
Author(s):  
Feng-Qun Zhao ◽  
Zhong-Min Wang
Keyword(s):  
Differential Equations ◽  
Cantilever Beam ◽  
Functionally Graded Material ◽  
Equations Of Motion ◽  
Admissible Function ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Cross Sectional ◽  
Graded Material ◽  
The Cross

Transverse vibration and stability of deploying rectangular cross-sectional cantilever beam made of functionally graded material are investigated. The functionally graded material beam is assumed to be constructed with ceramics and metal phases, and the corresponding equivalent parameters of functionally graded material are found to continuously vary across the cross-sectional height with a simple power law. The differential equations of motion of deploying functionally graded material cantilever beam are derived by Hamilton’s principle. Based on the assumed modal method, the beam deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. The eigenfunctions of cantilever beam in which the length of the beam is time-dependent are chosen as admissible functions. Galerkin method is employed to discretize the differential equation to a set of time-coordinate-dependent ordinary differential equations, and then the eigenvalue problem depending on time is obtained. The deployment motion of beam is considered as a constant speed in this study, and some numerical results, which are variations of tip deflection response and complex frequencies, are obtained. Finally, the effects of gradient index of functionally graded material, deploying speed, initial length and protruded length, the cross-sectional height on dynamical characteristics, and divergence instability of the deploying functionally graded material beam are discussed.

Application of radial basis function-based meshless method for torsional analysis of elliptical and bone shaped-irregular sections

2021 ◽  
pp. 146442072110534
Author(s):  
Ram Bilas Prasad ◽  
Jeeoot Singh ◽  
Karunesh Kumar Shukla
Keyword(s):  
Differential Equations ◽  
Radial Basis Function ◽  
Functionally Graded Material ◽  
Basis Function ◽  
Meshless Method ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Radial Basis ◽  
Graded Material ◽  
Torsional Analysis

This article presents a torsional analysis of solid elliptical, hollow circular, and actual bone sections of orthotropic and functionally graded material. The formulation of the governing equation is done using the Saint-Venant torsion theory. A classical power law is considered for the modelling of functionally graded material. Five different radial basis functions-based meshless methods are used for the discretization of the governing differential equations. MATLAB code is developed to solve the discretized partial differential equations. A convergence and validation study has been carried out to demonstrate the effectiveness and accuracy of the present method. Numerical examples for torsional rigidity and shear stresses are presented for circular, elliptical, and bone-shaped irregular sections made up of orthotropic and functionally graded materials. Finally, the proposed radial basis function-based meshless method is applied to the modelling and torsional analysis of an actual bone cross-section.

Analytical Solution for the Thermopiezoelastic Behavior of a Smart Functionally Graded Material Hollow Sphere Under Radially Symmetric Loadings

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
M. Jabbari ◽  
A. R. Barati
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Hollow Sphere ◽  
Material Properties ◽  
Functionally Graded ◽  
Feedback Gain ◽  
Modern Engineering ◽  
Graded Material ◽  
Power Law Function ◽  
Fgm Layer

An analytical study of the piezothermoelastic behavior of a functionally graded material (FGM) hollow sphere with integrated piezoelectric layers as a sensor and actuator under the effect of radially symmetric thermo-electro-mechanical loading is carried out. The material properties of the FGM layer are assumed to be graded in the radial direction according to a power law function. Governing differential equations are developed in terms of the components of the displacement field, the electric potential and the temperature of each layer of the smart FGM hollow sphere. The resulting differential equations are solved analytically. Numerical examples are given and discussed to show the significant influence of grading index of material properties and feedback gain on the mechanical–electrical responses. This will be useful for modern engineering design.

Nonlinear Vibration of Functionally Graded Material Cylindrical Shell Based on Reddy’s Third-Order Plates and Shells Theory

2012 ◽  
Vol 625 ◽  
pp. 18-24 ◽  
Author(s):  
Lu Dong ◽  
Yu Xin Hao ◽  
Jian Hua Wang ◽  
Li Yang
Keyword(s):  
Cylindrical Shell ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Third Order ◽  
Partial Differential ◽  
Graded Material ◽  
Plates And Shells

In this paper, an analysis on nonlinear dynamics of a simply supported functionally graded material (FGM) cylindrical shell subjected to the different excitation in thermal environment. Material properties of cylindrical shell are assumed to be temperature-dependent. Based on the Reddy’s third-order plates and shells theory[1], the nonlinear governing partial differential equations of motion for the FGM cylindrical shell are derived by using Hamilton’s principle. Galerkin’s method is utilized to transform the partial differential equations into a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitation. The effects played by different excitation and system initial conditions on the nonlinear vibration of the cylindrical shell are studied. In addition, the Runge–Kutta method is used to find out the nonlinear dynamic responses of the FGM cylindrical shell.

Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect

2015 ◽  
Vol 18 (2) ◽  
pp. 425-442 ◽  
Author(s):  
Fouzia Larbi Chaht ◽  
Abdelhakim Kaci ◽  
Mohammed Sid Ahmed Houari ◽  
Abdelouahed Tounsi ◽  
O. Anwar Beg ◽  
...  
Keyword(s):  
Functionally Graded Material ◽  
Functionally Graded ◽  
Size Dependent ◽  
Graded Material ◽  
Stretching Effect

scholarly journals Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Finite Element ◽  
Static Analysis ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Nonlinear Static Analysis ◽  
Geometrically Nonlinear ◽  
Timoshenko Beams ◽  
Graded Material ◽  
Nonlinear Static ◽  
Fgm Beam

Geometrically nonlinear static analysis of edge cracked cantilever Timoshenko beams composed of functionally graded material (FGM) subjected to a nonfollower transversal point load at the free end of the beam is studied with large displacements and large rotations. Material properties of the beam change in the height direction according to exponential distributions. The cracked beam is modeled as an assembly of two subbeams connected through a massless elastic rotational spring. In the study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. The convergence study is performed for various numbers of finite elements. In the study, the effects of the location of crack, the depth of the crack, and various material distributions on the nonlinear static response of the FGM beam are investigated in detail. Also, the difference between the geometrically linear and nonlinear analysis of edge cracked FGM beam is investigated in detail.

Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion

2017 ◽  
Vol 232 (6) ◽  
pp. 930-940 ◽  
Author(s):  
B Panigrahi ◽  
G Pohit
Keyword(s):  
Centrifugal Force ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Ritz Method ◽  
Functionally Graded ◽  
Neutral Axis ◽  
Spanwise Direction ◽  
Governing Equations ◽  
Graded Material ◽  
Fixed End

This study investigates the stiffening effect due to rotation on the nonlinear vibrational characteristics for cracked Timoshenko beam for the first time. Fixed end of the beam is attached to a rotating hub. Functionally graded material is taken into consideration, in which the properties vary as a continuous function along the depth of the beam. An elastic mass-less rotational spring is assumed in the place of crack, which splits the beam into two different parts. The point on the neutral axis at the fixed end is assumed to be the center of rotation of the beam. Centrifugal force is considered to act towards the spanwise direction and along the neutral axis. An additional displacement due to rotation of the beam along with the centrifugal force is incorporated with the energy formulation. Timoshenko beam theory and classical Ritz method is employed to derive the governing equations. In order to solve the nonlinear governing equations, direct substitution iterative technique is used. Effects of various parameters such as rotating speeds, radius of hub, depth of crack, location of crack, and different functionally graded material properties on linear and nonlinear vibration characteristics are studied. Validity of the present methodology is assured by comparing the results with some of the results from the existing literatures.

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