Theories and Analysis of Functionally Graded Beams

Reddy, J. N.; Ruocco, Eugenio; Loya, J.A.; Neves, A. M. A.

doi:10.3390/app11157159

Cited by 16 publications

(4 citation statements)

References 40 publications

(49 reference statements)

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“…Similar but simpler beams can be found in the literature [7,11,12]. Numerous works have been separately dedicated to the mechanics of FGM beams [13,14], non-uniform beams [15,16], beams on variable elastic foundations [17,18], and beams subjected to non-uniformly distributed loads, but have rarely considered these factors simultaneously. There are many works on axially functionally graded (AFG) beams with variable cross-sections, almost focusing on solving their natural frequencies and mode shapes using various analytical [14,19] or numerical methods [12,20,21], with only a few exceptions that considered dynamic responses.…”

Section: Introductionmentioning

confidence: 91%

A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

Liu,

Huang,

Zhao

2023

Aerospace

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Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated.

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Section: Introductionmentioning

confidence: 91%

A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

Liu,

Huang,

Zhao

2023

Aerospace

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“…Furthermore, the third shear deflection beam theory was examined by Gangnian et al [7] for the nonlinear bending of FG beams, and the differential quadrature method was utilized to obtain numerical results. Reddy et al [8] studied the classical first-order and third-order shear deformation of FG straight beams, and analytical solutions for bending were determined.…”

Section: Introductionmentioning

confidence: 99%

Longitudinal–Transverse Vibration of a Functionally Graded Nanobeam Subjected to Mechanical Impact and Electromagnetic Actuation

Herişanu

Marinca

2023

Symmetry

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This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal–transverse vibration of a simple supported Euler–Bernoulli beam, which accounted for von Kármán geometric nonlinearity, including the first-order strain–displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler–Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton’s principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal–transverse governing equations were discretized using the Galerkin–Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh–Hurwitz criteria, and global stability was studied using the Lyapunov function.

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“…Reddy et al [13] published a review paper about the governing equations and analytical solutions of the classical and shear deformation theories of FGM beams. More specifically, the classical, first-order, and third-order shear deformation theories accounted for the through thickness variation in a dual-phase FGM and modified couple stress (i.e., strain gradient) and von Kármán nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Static and Free Vibration Analyses of Functionally Graded Plane Structures

Gaspar,

Loja,

Barbosa

2023

J. Compos. Sci.

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In recent years, the use of functionally graded materials has been the focus of several studies due to their intrinsic ability to be tailored according to the requirements of structures while minimising abrupt stress transitions commonly found in laminated composites. In most studies, the materials’ mixture gradient is established through a structural component, i.e., thickness, which is known to visibly enhance structural behaviour. However, depending on the type of structure, it is important to exploit the possibility of building a structure using other gradient directions. The innovative characteristic of this work, which aims to study plane truss and frame-type structures made of functionally graded materials, lies in the specificity that the materials’ mixture gradient occurs as a function of a geometric structure feature, i.e., for example, the structure height, rather than the more usual approach, as a component dependence, i.e., through a member thickness or even along its length. The performance of the present model is illustrated through a set of case studies, and where possible, the results achieved are compared with more traditional solutions.

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Theories and Analysis of Functionally Graded Beams

Cited by 16 publications

References 40 publications

A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

Longitudinal–Transverse Vibration of a Functionally Graded Nanobeam Subjected to Mechanical Impact and Electromagnetic Actuation

Static and Free Vibration Analyses of Functionally Graded Plane Structures

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