Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

2019 ◽  
Vol 48 (4) ◽  
pp. 459-479 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Hai Thanh Tran ◽  
Do Nam
Keyword(s):  
Modal Analysis ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
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.

On modal analysis of axially functionally graded material beam under hygrothermal effect

2019 ◽  
Vol 234 (5) ◽  
pp. 1085-1101 ◽  
Author(s):  
Pankaj Sharma ◽  
Rahul Singh ◽  
Muzamal Hussain
Keyword(s):  
Modal Analysis ◽  
Functionally Graded Material ◽  
Volume Fraction ◽  
Axial Direction ◽  
Functionally Graded ◽  
Element Analysis ◽  
Hygrothermal Effect ◽  
Graded Material ◽  
Eigen Frequencies ◽  
Axially Functionally Graded Material

This investigation focuses on the modal analysis of an axially functionally graded material beam under hygrothermal effect. The material constants of the beam are supposed to be graded smoothly along the axial direction under both power law and sigmoid law distribution. A finite element analysis with COMSOL Multiphysics® (version 5.2) package is used to find the Eigen frequencies of the beam. The accuracy of the technique is authenticated by relating the results with the prior investigation for reduced case. The effects of moisture changes, temperature, and volume fraction index, length-to-thickness ratio on the Eigen frequencies are investigated in detail. It is believed that the present investigation may be useful in the design of highly efficient environmental sensors for structural health monitoring perspective.

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.

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.

Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material

2011 ◽  
Vol 18 (1-2) ◽  
pp. 21-34 ◽  
Author(s):  
Turgut Kocatürk ◽  
Mesut Şimşek ◽  
Şeref Doğuşcan Akbaş
Keyword(s):  
Finite Element ◽  
Static Analysis ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Element Approximation ◽  
Large Rotations ◽  
Non Linear ◽  
Graded Material ◽  
Newton Raphson

AbstractIn this study, non-linear static analysis of a cantilever Timoshenko beam composed of functionally graded material (FGM) under a non-follower transversal uniformly distributed load is studied with large displacements and large rotations. Material properties of the beam change in the thickness direction according to a power-law function. In this study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The non-linear problem is solved by using the incremental displacement-based finite element method in conjunction with the Newton-Raphson iteration method. To use the solution procedures of the Newton-Raphson method, there is a need to linearize equilibrium equations, which can be achieved through the linearization of the principle of virtual work. In this study, the effects of large deflections, large rotations and various material distributions on displacements and normal stress and shear stress distributions through the thickness of the beam are investigated in detail. The convergence study is performed for various numbers of finite elements. In addition, some of the particular results of the present study which are obtained for the homogeneous material case are compared with the results of the SAP2000 packet program. Numerical results show that geometrical non-linearity and material distribution play very important roles in the static responses of FGM beams.

Sign in / Sign up

Export Citation Format

Share Document