On Free Vibration of Functionally Graded Euler-Bernoulli Beam Models Based on the Non-Local Theory

2012 ◽  
Author(s):  
Reza Moheimani ◽  
M. T. Ahmadian
Keyword(s):  
Free Vibration ◽  
Axial Load ◽  
Power Index ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Local Theory ◽  
Local Parameter ◽  
Bernoulli Beam ◽  
Non Local ◽  
Euler Bernoulli Beam

In this paper, the governing equations and boundary conditions of a functionally graded Euler-Bernoulli beam are developed based on the non-local theory of elasticity. Afterward, the free vibration is investigated and the effects of the axial load, the non-local parameter and the power index on the natural frequency of a hinged-hinged beam is assessed. The results indicate that the non-local parameter has a decreasing effect on the frequency while the power index has an increasing effect. It is also noted that the effect of the axial load is increasing too.

scholarly journals Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory

Vibration ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 51-69
Author(s):  
Reza Moheimani ◽  
Hamid Dalir
Keyword(s):  
Power Index ◽  
Linear Equations ◽  
Local Effect ◽  
Functionally Graded ◽  
Local Theory ◽  
Classical Elasticity ◽  
Graded Materials ◽  
Nano Scale ◽  
Non Local ◽  
External Loads

Functionally graded materials (FGMs) have wide applications in different branches of engineering such as aerospace, mechanics, and biomechanics. Investigation of the mechanical behaviors of structures made of these materials has been performed widely using classical elasticity theories in micro/nano scale. In this research, static, dynamic and vibrational behaviors of functional micro and nanobeams were investigated using non-local theory. Governing linear equations of the problem were driven using non-local theory and solved using an analytical method for different boundary conditions. Effects of the axial load, the non-local parameter and the power index on the natural frequency of different boundary condition are assessed. Then, the obtained results were compared with those obtained from classical theory. These results showed that a non-local effect could greatly affect the behaviors of these beams, especially at nano scale.

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 Effect of Axial Force on the Free Vibration of an Euler-Bernoulli Beam Carrying a Number of Various Concentrated Elements

2013 ◽  
Vol 20 (3) ◽  
pp. 357-367 ◽  
Author(s):  
Gürkan Şcedilakar
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Axial Load ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Bernoulli Beam ◽  
Euler Beam ◽  
Euler Bernoulli Beam

In this study, free vibration analysis of beams carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring-mass systems subjected to the axial load was performed. All analyses were performed using an Euler beam assumption and the Finite Element Method. The beam used in the analyses is accepted as pinned-pinned. The axial load applied to the beam from the free ends is either compressive or tensile. The effects of parameters such as the number of spring-mass systems on the beam, their locations and the axial load on the natural frequencies were investigated. The mode shapes of beams under axial load were also obtained.

scholarly journals Free vibration of rectangular nanoplate strips

2019 ◽  
Author(s):  
Jaan Lellep ◽  
Mainul Hossain
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Theory Of Elasticity ◽  
Local Theory ◽  
Natural Vibrations ◽  
Size Dependent ◽  
Nano Structures ◽  
Non Local ◽  
Local Compliance

Natural vibrations of nanobeams and nanosheets are investigated with the help of nonlocal theories of elasticity. The vibration analysis is based on the size-dependent non-local theory of elasticity developed by A. C. Eringen. It is assumed that the nano-structures under consideration have rectangular cross sections with piece wise constant dimensions and that the nanoplates are weakened with defects. The influence of the crack on the vibration of the nanoplate is assessed with the aid of additional local compliance developed in previous papers. Numerical results are presented for one- and two-stepped nanoplates.

Study of internal stresses in rock massif within the framework of elastic layered block models with inclusions of the hierarchical structure of the L-rank.

2021 ◽  
Author(s):  
Olga Hachay ◽  
Andrey Khachay
Keyword(s):  
Hierarchical Structure ◽  
Classical Theory ◽  
Acoustic Waves ◽  
Degrees Of Freedom ◽  
Heterogeneous Media ◽  
Theory Of Elasticity ◽  
Rock Massif ◽  
Local Theory ◽  
Internal Stresses ◽  
Non Local

<p>In recent years, new models of continuum mechanics, generalizing the classical theory of elasticity, have been intensively developed. These models are used to describe composite and statistically heterogeneous media, new structural materials, as well as in complex massifs in mine conditions. The paper presents an algorithm for the propagation of longitudinal acoustic waves in the framework of active well monitoring of elastic layered block media with inclusions of hierarchical type of L-th rank. Relations for internal stresses and strains for each hierarchical rank are obtained, which constitute the non local theory of elasticity. The essential differences between the non local theory of elasticity and the classical one and the connection between them are investigated. A characteristic feature of the theory of media with a hierarchical structure is the presence of scale parameters in explicit or implicit form. This work focuses on the study of the effects of non locality and internal degrees of freedom, reflected in internal stresses, which are not described by the classical theory of elasticity and which can be potential precursors of the development of a catastrophic process in a rock massif. Thanks to the use of a model of a layered block medium with hierarchical inclusions, it is possible, using borehole acoustic monitoring, to determine the position of the highest values ​​of internal stresses and, with less effort, to implement the method of unloading the rock massif. If it is necessary to conduct short-term predictive monitoring of geodynamic regions and determine a more accurate position of the source of a dynamic phenomenon using borehole active acoustic observations, it is necessary to use the values ​​of the tensor of internal hierarchical stresses as a monitored parameter.</p>

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