Analytical responses of functionally graded beam under moving mass using Caputo and Caputo–Fabrizio fractional derivative models

2020 ◽  
Vol 26 (19-20) ◽  
pp. 1859-1867 ◽  
Author(s):  
Ibrahim M Abu-Alshaikh ◽  
Amro A Almbaidin
Keyword(s):  
Fractional Derivative ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Moving Mass ◽  
Functionally Graded Beam ◽  
The Laplace Transform ◽  
Voigt Model ◽  
The Difference ◽  
Euler Bernoulli Beam

In this article, a functionally graded simply supported Euler–Bernoulli beam subjected to moving mass is considered in which the beam-damping is described using fractional Kelvin–Voigt model. A comparison between Caputo and Caputo–Fabrizio fractional derivatives for obtaining the analytical dynamic response of the beam is carried out. The equation of motion is solved by the decomposition method with the cooperation of the Laplace transform. Two verification studies were performed to check the validity of the solutions. The results show that the grading order, the velocity of the moving mass and the fractional derivative order have significant effects on the beam deflection, whereas the difference between the results of the two fractional derivative models is expressed by the determination of the correlation coefficient.

scholarly journals Composite patch reinforcement of a cracked simply-supported beam traversed by moving mass

2020 ◽  
Vol 14 (1) ◽  
pp. 6403-6415
Author(s):  
M. S. Aldlemy ◽  
S. A. K. Al-jumaili ◽  
R. A. M. Al-Mamoori ◽  
T. Ya ◽  
Reza Alebrahim
Keyword(s):  
Finite Element Method ◽  
Edge Crack ◽  
Time History ◽  
Beam Theory ◽  
Beam Deflection ◽  
Moving Mass ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Composite Patch ◽  
Euler Bernoulli Beam

In this study dynamic analysis of a metallic beam under travelling mass was investigated. A beam with an edge crack was considered to be reinforced using composite patch. Euler-Bernoulli beam theory was applied to simulate the time-history behavior of the beam under dynamic loading. Crack in the beam was modeled using a rotational spring. Dimension of the composite patch, crack length, stress intensity factor at crack tip and beam deflection are some parameters which were studied in details. Results were validated against those which were found through Finite Element Method.

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.

scholarly journals Prediction of initial shape of functionally graded ceramic pre-forms for near-net-shape sintering

2003 ◽  
Vol 35 (1) ◽  
pp. 5-12 ◽  
Author(s):  
A.L. Maximenko ◽  
Der Biest ◽  
E.A. Olevsky
Keyword(s):  
Numerical Procedure ◽  
Continuum Theory ◽  
Functionally Graded ◽  
Theoretical Determination ◽  
Initial Shape ◽  
Ceramic Components ◽  
Near Net Shape ◽  
The Difference ◽  
The Continuum

Sintering of macroscopically inhomogeneous ceramic components is always accompanied by shape distortions due to the difference in shrinkage rates of powder elements. The objective of the present investigation is the theoretical determination of initially distorted shapes of the green bodies needed to provide near-net-shape sintered components. Calculations are based on a finite element implementation of the continuum theory of sintering. In this theory, sintering is considered as a creep under the influence of the compressive ?sintering stress?. To predict the initial shapes of the components the ?inverse? numerical procedure is used when the component is assumed to swell from the final shape to the initial one under the influence of the pressure equal (with the exception of sign) to the sintering stress.

Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

2011 ◽  
Vol 189-193 ◽  
pp. 3391-3394 ◽  
Author(s):  
Qing Zhao Yao ◽  
Lin Chao Liu ◽  
Qi Fang Yan
Keyword(s):  
Fractional Derivative ◽  
Mechanical Behavior ◽  
Constitutive Relations ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Bernoulli Beam ◽  
The Laplace Transform ◽  
Static Mechanical ◽  
Euler Bernoulli Beam

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation

2020 ◽  
Vol 12 (08) ◽  
pp. 2050093
Author(s):  
Mehdi Mousavi Khoram ◽  
Mohammad Hosseini ◽  
Amin Hadi ◽  
Mohammad Shishehsaz
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Pasternak Foundation ◽  
Functionally Graded Beam ◽  
Aspect Ratios ◽  
The Difference ◽  
Generalized Differential

Bending of bidirectional functionally graded nanobeams under mechanical loads and magnetic force was investigated. The nanobeam is assumed to be resting on the Winkler–Pasternak foundation. Eringen’s nonlocal elasticity theory and Timoshenko beam model are utilized to describe the mechanical behavior of the nanobeam. Material properties of the functionally graded beam are assumed to vary in the thickness and length of the nanobeam. Hamilton’s principle is employed to derive the governing equation and related boundary conditions. These equations are solved using the generalized differential quadrature method. The obtained results are compared with the results presented in other studies, to ensure the validity and versatility of this method. This comparison shows a good agreement between the results. Results are presented and discussed for different values of functionally graded materials indices, different aspect ratios, and different boundary conditions. The effect of the magnetic field and elastic foundation on buckling load has also been studied. The difference in nanobeam behavior for different values of the size-effect parameter is clearly shown.

Study on Natural Frequencies of Transverse Free Vibration of Functionally Graded Axis Beams by the Differential Quadrature Method

2019 ◽  
Vol 105 (6) ◽  
pp. 1095-1104
Author(s):  
Jin-lun Zhang ◽  
Liao-jun Zhang ◽  
Ren-yu Ge ◽  
Li Yang ◽  
Jun-wu Xia
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Functionally Graded ◽  
Differential Quadrature ◽  
Variable Cross ◽  
Quadrature Method ◽  
Bernoulli Beam ◽  
Functionally Graded Beam ◽  
Euler Bernoulli Beam

Functionally gradient materials with special mechanical characteristics are more and more widely used in engineering. The functionally graded beam is one of the commonly used components to bear forces in the structure. Accurate analysis of the dynamic characteristics of the axially functionally graded (AFG) beam plays a vital role in the design and safe operation of the whole structure. Based on the Euler-Bernoulli beam theory (EBT), the characteristic equation of transverse free vibration for the AFG Euler-Bernoulli beam with variable cross-section is obtained in the present work, and the governing equations of the beam are transformed into ordinary differential equations with variable coefficients. Using differential quadrature method (DQM), the solution formulas of characteristic equations under different boundary conditions are derived, and the natural frequencies of the AFG beam are calculated, while the node partition of a non-uniform geometric progression is discussed.

scholarly journals The nonlinear thermomechanical vibration of a functionally graded beam on Winkler-Pasternak foundation

2018 ◽  
Vol 148 ◽  
pp. 13004 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Geometric Nonlinearity ◽  
Beam Theory ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Bernoulli Beam ◽  
Functionally Graded Beam ◽  
Pasternak Elastic Foundation ◽  
Euler Bernoulli Beam

By using the Optimal Auxiliary Functions Method (OAFM), nonlinear free thermomechanical vibration of functionally graded beam (FGB) on Winkler-Pasternak elastic foundation is studied. Based on von Karman geometric nonlinearity, on Euler-Bernoulli beam theory and also on Galerkin procedure we obtain a second-order nonlinear differential equation with quadratic and cubic nonlinear terms. The results obtained by means of OAFM are compared and shown to be in an excellent agreement with available solutions known in the literature.

scholarly journals A Simplified Stress Analysis of Functionally Graded Beams and Influence of Material Function on Deflection

2021 ◽  
Vol 11 (24) ◽  
pp. 11747
Author(s):  
Fadi Althoey ◽  
Elias Ali
Keyword(s):  
Shear Stress ◽  
Stress Analysis ◽  
Elastic Moduli ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Material Function ◽  
Functionally Graded Beam ◽  
Structural Applications ◽  
Beam Thickness ◽  
Fgm Beam

This paper aims at providing a simplified analytical solution for functionally graded beam stress analysis and optimized material gradation on the beam deflection. The power-law (P-FGM) and exponential (E-FGM) material functions were considered for an exact solution of the normal and shear stress distributions across the beam thickness. Optimization of material function on the FGM beam deflection, which is new of its kind, was also investigated considering both simply supported and cantilever beams. It was observed that the non-dimensional normal stress and shear stress are independent of the elastic moduli values of the constituent materials but rather depends on both the ratio of the elastic moduli and the location across the beam thickness in the E-FGM material function model. This observation was first validated from available kinds of literature and through numerical simulation using ABAQUS and extended to the P-FGM stress analysis. The maximum deflection on the FGM beam occurred for a homogenous steel beam while the minimum deflection was observed on the beam with a P-FGM material function. The results of this work demonstrate that if properly designed and optimized, FGMs can provide an alternative material solution in structural applications.

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