Extension of Boley’s method to functionally graded beams

Element Analysis ◽

Perfect Agreement ◽

AbstractThe objective of this contribution is the computation of the Airy stress function for functionally graded beam-type structures subjected to transverse and shear loads. For simplification, the material parameters are kept constant in the axial direction and vary only in the thickness direction. The proposed method can be easily extended to material varying in the axial and thickness direction. In the first part an iterative procedure is applied for the determination of the stress function by means of Boley’s method. This method was successfully applied by Boley for two-dimensional (2D) isotropic plates under plane stress conditions in order to compute the stress distribution and the displacement field. In the second part, a shear loaded cantilever made of isotropic, functionally graded material is studied in order to verify our theory with finite element results. It is assumed that the Young’s modulus varies exponentially in the transverse direction and the Poisson ratio is constant. Stresses and displacements are analytically determined by applying our derived theory. Results are compared to a 2D finite element analysis performed with the commercial software ABAQUS. It is found that the analytical and numerical results are in perfect agreement.

Finite element analysis of flexible functionally graded beams with variable Poisson’s ratio

Engineering Computations ◽

10.1108/ec-08-2015-0225 ◽

2016 ◽

Vol 33 (8) ◽

pp. 2421-2447 ◽

Cited By ~ 6

Author(s):

João Paulo Pascon

Keyword(s):

Finite Element Analysis ◽

Finite Element ◽

Poisson’S Ratio ◽

Scientific Literature ◽

Functionally Graded ◽

Poisson's Ratio ◽

Transverse Strain ◽

Thickness Direction ◽

Element Analysis ◽

Content Type

Purpose The purpose of this paper is to deal with large deformation analysis of plane beams composed of functionally graded (FG) elastic material with a variable Poisson’s ratio. Design/methodology/approach The material is assumed to be linear elastic, with a Poisson’s ratio varying according to a power law along the thickness direction. The finite element used is a plane beam of any-order of approximation along the axis, and with four transverse enrichment schemes, which can describe constant, linear, quadratic and cubic variation of the strain along the thickness direction. Regarding the constitutive law, five materials are adopted: two homogeneous limiting cases, and three intermediate FG cases. The effect of both finite element kinematics and distribution of Poisson’s ratio on the mechanical response of a cantilever is investigated. Findings In accordance with the scientific literature, the second scheme, in which the transverse strain is linearly variable, is sufficient for homogeneous long (or thin) beams under bending. However, for FG short (or moderate thick) beams, the third scheme, in which the transverse strain variation is quadratic, is needed for a reliable strain or stress distribution. Originality/value In the scientific literature, there are several studies regarding nonlinear analysis of functionally graded materials (FGMs) via finite elements, analysis of FGMs with constant Poisson’s ratio, and geometrically linear problems with gradually variable Poisson’s ratio. However, very few deal with finite element analysis of flexible beams with gradually variable Poisson’s ratio. In the present study, a reliable formulation for such beams is presented.

Effects of piezoelectric rings on electro-elasto-dynamic behaviour of functionally graded cylindrical shell

Journal of Intelligent Material Systems and Structures ◽

10.1177/1045389x16651154 ◽

2016 ◽

Vol 28 (2) ◽

pp. 272-289 ◽

Cited By ~ 1

Author(s):

Mohammadreza Saviz

Keyword(s):

Finite Element ◽

Cylindrical Shell ◽

Volume Fraction ◽

Virtual Work ◽

Electrical Potential ◽

Axial Direction ◽

Functionally Graded ◽

Radial Coordinate ◽

A layer-wise finite element approach is adopted to analyse the hollow cylindrical shell made of functionally graded material with piezoelectric rings as sensor/actuator, under dynamic load. The mechanical properties of the substrate are regulated by volume fraction as a function of radial coordinate. The thickness of functionally graded material shell and piezo-rings is divided into mathematical sub-layers and then the general layer-wise laminate theory is formulated through introducing piecewise continuous approximations across the thickness, accounting for any discontinuity in derivatives of the displacement at the interface between the ring and cylinder. The virtual work statement including structural and electrical potential energies yields the three-dimensional governing equations which are reduced to two-dimensional differential equations, using layer-wise method. For axisymmetric case, the resulted equations are solved with one-dimensional finite element method in the axial direction. By assembling stiffness and mass matrices, the required stress and displacement continuities at each interface and between the two adjacent elements are forced. The results for free vibration and static loading are applied to study the convergence and verified by comparing them to solutions of similar existing problems. The induced deformation by piezoelectric actuators as well as the effect of rings on functionally graded material shell is investigated.

Finite Element Analysis of Human Tibia Modeled as a Functionally Graded Material

Journal of Engineering and Science in Medical Diagnostics and Therapy ◽

10.1115/1.4044054 ◽

2019 ◽

Vol 2 (3) ◽

Author(s):

Ashish Tiwari ◽

Pankaj Wahi ◽

Niraj Sinha

Keyword(s):

Finite Element Analysis ◽

Finite Element ◽

Stress Distribution ◽

Material Properties ◽

Functionally Graded ◽

Element Analysis ◽

Cross Sectional ◽

Human Tibia ◽

Human tibia, the second largest bone in human body, is made of complex biological material having inhomogeneity and anisotropy in such a manner that makes it a functionally graded material. While analyses of human tibia assuming it to be made of different material regions have been attempted in past, functionally graded nature of the bone in the mechanical analysis has not been considered. This study highlights the importance of functional grading of material properties in capturing the correct stress distribution from the finite element analysis (FEA) of human tibia under static loading. Isotropic and orthotropic material properties of different regions of human tibia have been graded functionally in three different manners and assigned to the tibia model. The nonfunctionally graded and functionally graded models of tibia have been compared with each other. It was observed that the model in which functional grading was not performed, uneven distribution and unrealistic spikes of stresses occurred at the interfaces of different material regions. On the contrary, the models with functional grading were free from this potential artifact. Hence, our analysis suggests that functional grading is essential for predicting the actual distribution of stresses in the entire bone, which is important for biomechanical analysis. We find that orthotropic nature of the bone tends to increase the maximum von Mises stress in the entire tibia, while inclusion of cross-sectional inhomogeneity typically increases the stresses across normal cross section. Accordingly, our analysis suggests that both orthotropy as well as cross-sectional inhomogeneity should be included to correctly capture the stress distribution in the bone.

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

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

10.1177/0954406219888234 ◽

2019 ◽

Vol 234 (5) ◽

pp. 1085-1101 ◽

Cited By ~ 6

Author(s):

Pankaj Sharma ◽

Rahul Singh ◽

Muzamal Hussain

Keyword(s):

Modal Analysis ◽

Axially Functionally Graded Material

Volume Fraction ◽

Axial Direction ◽

Functionally Graded ◽

Element Analysis ◽

Hygrothermal Effect ◽

Graded Material ◽

Eigen Frequencies ◽

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.

Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing ◽

Non-Linear Finite Element Analysis of Functionally Graded Material Sector Plates

10.4203/ccp.77.32 ◽

2009 ◽

Author(s):

M. Salehi ◽

M. Tayefeh

Keyword(s):

Finite Element Analysis ◽

Finite Element ◽

Functionally Graded ◽

Element Analysis ◽

Linear Finite Element ◽

Non Linear ◽

A Comparison of Finite Element Beam Formulations for Thermoelastic Behavior of Functionally Graded Structures

Aerospace ◽

10.1115/imece2005-80999 ◽

2005 ◽

Author(s):

Thomas G. Eason ◽

Simo´n A. Caraballo

Keyword(s):

Finite Element ◽

Shear Deformation Theory ◽

Axial Direction ◽

Beam Element ◽

Functionally Graded ◽

Shell Elements ◽

Static Condensation ◽

Graded Structures ◽

Structural Scale

Modeling at the structural scale most often requires the use of beam and shell elements. This paper compares two finite element formulations based on first-order shear deformation theory undergoing thermo-mechanical loading. One formulation is a two node beam element employing static condensation based on the work of Chakraborty et al. The second formulation follows a more traditional route using FSOD theory for a three node beam element. Both formulations are used to investigate the behavior of a functionally graded beam under axial and through-the-thickness temperature gradients. Both formulations work well for a constant uniform mechanical or temperature loading. However, for beam structures containing a thermal gradient in the axial direction, the two node beam element performs poorly as compared to the three node element in terms of transverse shearing stress calculated from the equilibrium equation.

Nonlinear bending and free vibration response of SUS316-Al2O3 functionally graded plasma sprayed beams: theoretical and experimental study

Journal of Vibration and Control ◽

10.1177/1077546316659422 ◽

2016 ◽

Vol 24 (6) ◽

pp. 1171-1184 ◽

Cited By ~ 3

Author(s):

Pravin Malik ◽

Ravikiran Kadoli

Keyword(s):

Finite Element ◽

Free Vibration ◽

Maximum Error ◽

Functionally Graded ◽

Nonlinear Finite Element ◽

Element Analysis ◽

Nonlinear Bending ◽

Plasma Sprayed ◽

3D Solid

Functionally graded SUS316-Al2O3 beams with ceramic content varying from 0 to 40% were prepared by a plasma spraying technique. Nonlinear finite element analysis was used to obtain the static deflection and free vibration of a clamped-free functionally graded beam. Von Kármán geometric nonlinearity and power law variation in material gradation through the beam thickness are considered in the analysis. The maximum error between the experimental and nonlinear finite element results for deflection is 6.68% and 14.31% on the fundamental frequency. Numerical results have also been attempted using ANSYS 3D solid element and they compare more closely with the experimental results.

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Applied Mathematics and Mechanics ◽

10.1007/s10483-020-2664-8 ◽

2020 ◽

Vol 41 (12) ◽

pp. 1787-1804

Author(s):

N. V. Viet ◽

W. Zaki ◽

Quan Wang

Keyword(s):

Finite Element Analysis ◽

Finite Element ◽

Free Vibration ◽

Material Properties ◽

Rapid Development ◽

Functionally Graded ◽

Mode Shapes ◽

Element Analysis ◽

AbstractAdvancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on one-dimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either Timoshenko or Euler-Bernoulli beam theories. Then, Hamilton’s principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam’s natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

Numerical and experimental analysis of Cu–Fe functionally graded beam subjected to tensile loading

10.1177/0954406220917696 ◽

2020 ◽

Vol 234 (19) ◽

pp. 3837-3845

Author(s):

Maryam Torabian ◽

Seyed Mohammad Reza Khalili

Keyword(s):

Finite Element ◽

Tensile Strength ◽

Functionally Graded Materials ◽

Tensile Loading ◽

Functionally Graded ◽

Tensile Behaviour ◽

Graded Materials ◽

Functionally graded materials are new types of composites with heterogeneous microstructure in which some particular physical and mechanical properties change continuously in the thickness direction. In this research, a five-layer copper–iron functionally graded material was fabricated by changing the composition of the layers in a stepwise function between copper and iron using powder metallurgy method. The effect of fabrication process on the microstructure and tensile strength of functionally graded beam was investigated by using two types of presses: uniaxial press and cold iso-static press. Microscopic studies demonstrated appropriate connections between the layers and particles. To achieve ultimate tensile strength and strain, functionally graded copper–iron specimens were tested in tensile loading. The stress–strain graphs obtained from the test showed enhancement in tensile strength of copper and iron functionally graded beam compared to pure copper and iron beams. Finally, a model of this functionally graded material was analysed in ABAQUS finite element code, and the results were verified by experimental tests. Therefore, the present finite element model would be useful to investigate tensile behaviour of functionally graded materials.