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.

Three-dimensional buckling analyses of cracked functionally graded material plates via the MLS-Ritz method

2019 ◽  
Vol 134 ◽  
pp. 189-202 ◽  
Author(s):  
C.S. Huang ◽  
H.T. Lee ◽  
P.Y. Li ◽  
K.C. Hu ◽  
C.W. Lan ◽  
...  
Keyword(s):  
Functionally Graded Material ◽  
Ritz Method ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Graded Material

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 Control of Composition by Centrifugal Force in Functionally Graded Material.

1992 ◽  
Vol 58 (556) ◽  
pp. 2472-2477 ◽  
Author(s):  
Yoshimi WATANABE ◽  
Yasuyoshi FUKUI ◽  
Kenji NAKANISHI ◽  
Yoshio TAKEDA ◽  
Noboru YAMANAKA
Keyword(s):  
Centrifugal Force ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Graded Material

A tangent stiffness–based approach to study free vibration of shear-deformable functionally graded material rotating beam through a geometrically non-linear analysis

2017 ◽  
Vol 52 (5) ◽  
pp. 310-332 ◽  
Author(s):  
Suman Pal ◽  
Debabrata Das
Keyword(s):  
Free Vibration ◽  
Functionally Graded Material ◽  
Thermal Loading ◽  
Linear Analysis ◽  
Functionally Graded ◽  
Rotating Beam ◽  
Governing Equations ◽  
Tangent Stiffness ◽  
Non Linear ◽  
Graded Material

An improved mathematical model to study the free vibration behavior of rotating functionally graded material beam is presented, considering non-linearity up to second order for the normal and transverse shear strains. The study is carried out considering thermal loading due to uniform temperature rise and using temperature-dependent material properties. Power law variation is assumed for through-thickness symmetric functional gradation of ceramic–metal functionally graded beam. The effects of shear deformation and rotary inertia are considered in the frame-work of Timoshenko beam theory. First, the rotating beam configuration under time-invariant centrifugal loading and thermal loading is obtained through a geometrically non-linear analysis, employing minimum total potential energy principle. Then, the free vibration analysis of the deformed beam is performed using the tangent stiffness of the deformed beam configuration, and employing Hamilton’s principle. The Coriolis effect is considered in the free vibration problem, and the governing equations are transformed to the state-space to obtain the eigenvalue problem. The solution of the governing equations is obtained following Ritz method. The validation is performed with the available results, and also with finite element software ANSYS. The analysis is carried out for clamped-free beam and for clamped–clamped beam with immovably clamped ends. The results for the first two modes of chord-wise and flap-wise vibration in non-dimensional speed-frequency plane are presented for different functionally graded material compositions, material profile parameters, root offset parameters and operating temperatures.

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

2020 ◽  
pp. 095440622093630
Author(s):  
LV Kurpa ◽  
TV Shmatko
Keyword(s):  
Power Law ◽  
Functionally Graded Material ◽  
Complex Shape ◽  
Ritz Method ◽  
Mathematical Formulation ◽  
Functionally Graded ◽  
Buckling Load ◽  
Shallow Shells ◽  
Graded Material ◽  
R Functions

The purpose of the paper is to study stability and free vibrations of laminated plates and shallow shells composed of functionally graded materials. The approach proposed incorporates the Ritz method and the R-functions theory. It is assumed that the shell consists of three layers and is loaded in the middle plane. The both cases of uniform as well as non-uniform load are possible. The power-law distribution in terms of volume fractions is applied to get effective material properties for the layers. These properties are calculated for different arrangements and thicknesses of the layers by the analytical formulae obtained in the paper. The mathematical formulation is carried out in framework of the first-order shear deformation theory. The proposed approach consists of two steps. The first step is to define the pre-buckling state by solving the respective elasticity problem. The critical buckling load and frequencies of functionally graded material shallow shells are determined in the second step. The highlight of the method proposed is that it can be used for vibration and buckling analysis of plates and shallow shells of complex shape. The numerical results for frequencies and buckling load of plates and shallow shells of complex shape and different curvatures are presented to demonstrate the potential of the method developed. Different functionally graded material plates and shallow shells composed of a mixture of metal and ceramics are studied. The effects of the power law index, boundary conditions, thickness of the core, and face sheet layers on the fundamental frequencies and critical loads are discussed in this paper. The main advantage of the method is that it provides an analytical representation of the unknown solution, which is important when solving nonlinear problems.

The Nonlinear Dynamic Response of Functionally Graded Material Flat Spherical Shells Subjected to Thermal Loading

2012 ◽  
Vol 549 ◽  
pp. 580-583
Author(s):  
Yao Dai ◽  
Xiao Chong ◽  
Lei Zhang ◽  
Hong Qian Chen
Keyword(s):  
Functionally Graded Material ◽  
Thermal Loading ◽  
Method Of Lines ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Spherical Shells ◽  
Nonlinear Dynamic Response ◽  
The Method Of Lines ◽  
Graded Material

The response of functionally graded material flat spherical shells subjected to thermal loading is studied using the method of lines. Based on the Kirchhoff straight normal hypothesis and Von Karman's geometrically nonlinear theory, the governing equations are obtained. A semi-analytical numerical method, viz. the method of lines is introduced. Then, the partial differential equations are transformed into ordinary differential ones. The numerical results of flat spherical shells are given and compared with ones of the finite element method. The effects of the material gradient parameters on the responses are discussed in details.

A study on non-linear post-buckling behavior of tapered Timoshenko beam made of functionally graded material under in-plane thermal loadings

2016 ◽  
Vol 52 (1) ◽  
pp. 45-56 ◽  
Author(s):  
Amlan Paul ◽  
Debabrata Das
Keyword(s):  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Temperature Rise ◽  
Functionally Graded ◽  
Buckling Load ◽  
Uniform Temperature ◽  
Buckling Behavior ◽  
Non Linear ◽  
Graded Material ◽  
Post Buckling

In the present work, the non-linear post-buckling load–deflection behavior of tapered functionally graded material beam is studied for different in-plane thermal loadings. Two different thermal loadings are considered. The first one is due to the uniform temperature rise and the second one is due to the steady-state heat conduction across the beam thickness leading to non-uniform temperature rise. The governing equations are derived using the principle of minimum total potential energy employing Timoshenko beam theory. The solution is obtained by approximating the displacement fields following Ritz method. Geometric non-linearity for large post-buckling behavior is considered using von Kármán type non-linear strain-displacement relationship. Stainless steel/silicon nitride functionally graded material beam is considered with temperature-dependent material properties. The validation of the present work is successfully performed using finite element software ANSYS and using the available result in the literature. The post-buckling load–deflection behavior in non-dimensional plane is presented for different taperness parameters and also for different volume fraction indices. Normalized transverse deflection fields are presented showing the shift of the point of maximum deflection for various deflection levels. The results are new of its kind and establish benchmark for studying non-linear thermo-mechanical behavior of tapered functionally graded material beam.

UNCERTAIN BUCKLING AND RELIABILITY ANALYSIS OF THE PIEZOELECTRIC FUNCTIONALLY GRADED CYLINDRICAL SHELLS BASED ON THE NONPROBABILISTIC CONVEX MODEL

2014 ◽  
Vol 11 (06) ◽  
pp. 1350080 ◽  
Author(s):  
R. G. BI ◽  
X. HAN ◽  
C. JIANG ◽  
Y. C. BAI ◽  
J. LIU
Keyword(s):  
Shear Deformation ◽  
Functionally Graded Material ◽  
Structural Parameters ◽  
Cylindrical Shells ◽  
Functionally Graded ◽  
Governing Equations ◽  
Convex Model ◽  
Buckling Loads ◽  
Graded Material ◽  
The One

The uncertain buckling and reliability of the laminated piezoelectric functionally graded material (FGM) cylindrical shells subjected to axially compressed loads are investigated in this research. Considering the shear deformation, the buckling governing equations of the piezoelectric FGM cylindrical shells are derived on the basis of Donnell assumptions. And then the nonprobabilistic convex model is introduced to predict the uncertain buckling loads of the piezoelectric FGM cylindrical shells resulting from the unavoidable scatter in structural parameters. Finally, the reliability degree of the structures is obtained by computing the ratio of the multidimensional volume falling into the reliability domain to the one of the whole convex model. Numerical results indicate that uncertainties in structural parameters have significant effects on the critical buckling loads and reliability of the piezoelectric FGM cylindrical shells.

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