Geometrically Nonlinear Free Vibration of Composite Materials: Clamped-Clamped Functionally Graded Beam with an Edge Crack Using Homogenisation Method

2017 ◽  
Vol 730 ◽  
pp. 521-526 ◽  
Author(s):  
Mohcine Chajdi ◽  
El Bekkaye Merrimi ◽  
Khalid El Bikri
Keyword(s):  
Free Vibration ◽  
Edge Crack ◽  
Volume Fraction ◽  
Crack Depth ◽  
Beam Theory ◽  
Functionally Graded ◽  
Geometrically Nonlinear ◽  
Nonlinear Free Vibration ◽  
Functionally Graded Beam ◽  
Euler Bernoulli Beam

The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.

Stochastic Nonlinear Free Vibration Analysis of Functionally Graded Beam Subjected to Thermal Loading with Random Material Properties

2013 ◽  
Vol 747 ◽  
pp. 551-554
Author(s):  
Niranjan L. Shegokar ◽  
Achchhe Lal
Keyword(s):  
Free Vibration ◽  
Material Properties ◽  
Thermal Loading ◽  
Volume Fraction ◽  
Perturbation Technique ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Nonlinear Free Vibration ◽  
Functionally Graded Beam

This paper deals with the stochastic nonlinear free vibration response of functionally graded materials (FGMs) beam subjected to thermal loadings with uncertain material properties subjected to uniform and nonuniform temperature changes with temperature independent (TID) and dependent (TD) material properties. System properties such as material properties of each constituents material and volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strains using modified C0 continuity. A direct iterative based nonlinear finite element method in conjunction with first order perturbation technique (FOPT) is used for FGMs beam to compute the second order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency. The present outlined approach has been validated with the results available in literatures and independent Monte Carlo simulation (MCS).

scholarly journals An Efficient Beam Element Based on Quasi-3D Theory for Static Bending Analysis of Functionally Graded Beams

Materials ◽  
2019 ◽  
Vol 12 (13) ◽  
pp. 2198 ◽  
Author(s):  
Hoang Nam Nguyen ◽  
Tran Thi Hong ◽  
Pham Van Vinh ◽  
Do Van Thom
Keyword(s):  
Transverse Shear ◽  
Volume Fraction ◽  
Beam Theory ◽  
Beam Element ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Functionally Graded Beam ◽  
Transverse Shear Stresses ◽  
Power Law Index ◽  
Transverse Shear Strains

In this paper, a 2-node beam element is developed based on Quasi-3D beam theory and mixed formulation for static bending of functionally graded (FG) beams. The transverse shear strains and stresses of the proposed beam element are parabolic distributions through the thickness of the beam and the transverse shear stresses on the top and bottom surfaces of the beam vanish. The proposed beam element is free of shear-looking without selective or reduced integration. The material properties of the functionally graded beam are assumed to vary according to the power-law index of the volume fraction of the constituents through the thickness of the beam. The numerical results of this study are compared with published results to illustrate the accuracy and convenience rate of the new beam element. The influence of some parametrics on the bending behavior of FGM beams is investigated.

Free vibration analysis of a single edge cracked symmetric functionally graded stepped beams

2020 ◽  
Vol 23 (16) ◽  
pp. 3415-3428
Author(s):  
Yusuf Cunedioglu ◽  
Shkelzen Shabani
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Single Edge ◽  
Functionally Graded Beam ◽  
Cross Sectional

Free vibration analysis of a single edge cracked multi-layered symmetric sandwich stepped Timoshenko beams, made of functionally graded materials, is studied using finite element method and linear elastic fracture mechanic theory. The cantilever functionally graded beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. It is assumed that material properties vary continuously, along the thickness direction according to the exponential and power laws. A developed MATLAB code is used to find the natural frequencies of three types of the stepped beam, concluding a good agreement with the known data from the literature, supported also by ANSYS software in data verification. In the study, the effects of the crack location, crack depth, power law gradient index, different material distributions, different stepped length, different cross-sectional geometries on natural frequencies and mode shapes are analysed in detail.

Large Amplitude Free Vibration of Nanobeams Subjected to Magnetic Field Based on Nonlocal Elasticity Theory

2015 ◽  
Vol 764-765 ◽  
pp. 1199-1203 ◽  
Author(s):  
Tai Ping Chang
Keyword(s):  
Magnetic Field ◽  
Free Vibration ◽  
Nonlocal Elasticity ◽  
Beam Theory ◽  
Nonlinear Frequency ◽  
Bernoulli Beam ◽  
Nonlinear Free Vibration ◽  
Vibration Behavior ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

In the present study, nonlinear free vibration behavior of nanobeam subjected to magnetic field is investigated based on Eringen's nonlocal elasticity and Euler–Bernoulli beam theory. The Hamilton's principle is adopted to derive the governing equations together with Euler–Bernoulli beam theory and the von-Kármán's nonlinear strain–displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam under magnetic field by using the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented. The effect of nonlocal parameter on the nonlinear frequency ratio is studied; furthermore, the effect of magnetic field on the nonlinear free vibration behavior of nanobeam is investigated.

Electromechanical impedance response of a cracked functionally graded beam with imperfectly bonded piezoelectric wafers

2011 ◽  
Vol 22 (16) ◽  
pp. 1899-1912 ◽  
Author(s):  
Wei Yan ◽  
J Wang ◽  
WQ Chen ◽  
WC Li
Keyword(s):  
Lead Zirconate Titanate ◽  
Crack Depth ◽  
Beam Theory ◽  
Lead Zirconate ◽  
Functionally Graded ◽  
Beam Model ◽  
Functionally Graded Beam ◽  
Electromechanical Impedance ◽  
Smart Beam ◽  
Piezoelectric Wafers

An analytical model of a cracked functionally graded beam with attached Lead Zirconate Titanate (PZT) actuator/sensors is proposed in the paper for structural health monitoring. In this model, the dynamic behavior of the piezoelectric patches is considered and a viscoelastic law is adopted to describe the bonding imperfection between the piezoelectric patches and the beam. A piecewisely homogeneous beam model is then employed to approximate the original inhomogeneous beam based on the Timoshenko beam theory. The crack in the beam is treated as a massless rotational spring. In order to develop the recursive formulations to reduce the dimension of the final equations in the method of reverberation-ray matrix (MRRM), new local scattering relations are established for this smart beam using a matrix reduction technique. An analytical expression of the electromechanical impedance (EMI) is derived based on the improved MRRM via the recursive formulations. Comparison with existent experimental results and those predicted by other methods, such as the conventional MRRM, the transfer matrix method (TMM), and the finite element method (FEM), is made to validate the proposed analysis. Furthermore, the effects of various parameters including the crack depth on the EMI signatures are highlighted.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

Free Vibration Analysis of Functionally Graded Beams Resting on Elastic Foundation in Thermal Environment

2016 ◽  
Vol 16 (07) ◽  
pp. 1550029 ◽  
Author(s):  
P. Zahedinejad
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Functionally Graded ◽  
Power Law Distribution ◽  
Various Boundary Conditions

The free vibration of functionally graded (FG) beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment is studied using the third-order shear deformation beam theory. The material properties are temperature-dependent and vary continuously through the thickness direction of the beam, based on a power-law distribution in terms of the volume fraction of the material constituents. In order to discretize the governing equations, the differential quadrature method (DQM) in conjunction with the Hamilton’s principle is adopted. The convergence of the method is demonstrated. In order to validate the results, comparisons are made with solutions available for the isotropic and FG beams. Through a comprehensive parametric study, the effect of various parameters involved on the FG beam was studied. It is concluded that the uniform temperature rise has more significant effect on the frequency parameters than the nonuniform case.

scholarly journals Geometrically nonlinear free and forced vibrations analysis of clamped-clamped functionally graded beams with multicracks

2018 ◽  
Vol 211 ◽  
pp. 02002 ◽  
Author(s):  
Mohcine Chajdi ◽  
Ahmed Adri ◽  
Khalid El bikri ◽  
Rhali Benamar
Keyword(s):  
Volume Fraction ◽  
Geometrical Nonlinearity ◽  
Beam Theory ◽  
Forced Vibrations ◽  
Functionally Graded ◽  
Geometrically Nonlinear ◽  
Cracked Beam ◽  
Free And Forced Vibrations ◽  
Wide Range ◽  
Rotational Spring Model

Geometrically nonlinear free and forced vibrations of clampedclamped Functionally Graded beams with multi-cracks, located at different positions, based on the equivalent rotational spring model of crack and the transfer matrix method for beams is investigated. The FG beam properties are supposed to vary continuously through the thickness direction. The theoretical model is based on the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. A homogenization procedure, taking into account the presence of the crack, is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. Upon assuming harmonic motion, the discretized expressions for the total strain and kinetic energies of the beam are derived, and through application of Hamilton’s principle and spectral analysis, the problem is reduced to a nonlinear algebraic system solved using an approximate explicit method developed previously (second formulation) to obtain numerically the FG multi-cracked beam nonlinear fundamental mode and the corresponding backbone curves for a wide range of vibration amplitudes. The numerical results presented show the effect of the number of cracks, the crack depths and locations, and the volume fraction on the beam nonlinear dynamic response.

Sign in / Sign up

Export Citation Format

Share Document