Clamped-simply Supported Research Articles

Nonlocal buckling analysis of double-walled carbon nanotubes using initial value method and matricant approach

In this study, the buckling behavior of double-walled carbon nanotubes (DWCNTs) is investigated using the initial value method (IVM) combined with the approximate transfer matrix (ATM) approach within the framework of nonlocal elasticity theory. The aim is to provide a method that best estimates the critical buckling loads with improved accuracy and computational efficiency. To validate the method, the critical buckling loads obtained without considering nonlocal effects were compared with results from the literature, showing a high degree of agreement. Various boundary conditions, including simply supported (S-S), clamped-simply supported (C-S), clamped-free (C-F), and clamped-clamped (C-C), have been considered for investigation. The effects of nonlocal parameters, aspect ratios, and boundary conditions on the critical buckling loads are analyzed. The findings emphasize the importance of accounting for these parameters in mechanical analyses to ensure accurate predictions. The proposed framework provides an efficient alternative to existing numerical methods and offers valuable benchmarks for future research on the mechanical behavior of nanostructures.

Thermoelastic damping and frequency shift of different micro-scale piezoelectro-magneto-thermoelastic beams

This work focuses on mathematically studying thermoelastic damping (TED) and frequency shift (FS) in micro-scale piezoelectro-magneto-thermoelastic (PEMT) composite beams composed of BaTiO3-CoFe2O4 combination. Pertaining to cutting-edge micro-technologies implemented in several engineering/scientific applications now-a-days, micro-scale doubly clamped (CC), doubly simply supported (SS), clamped-free (CF), and clamped-simply supported (CS) beams are extensively analyzed. The beams are modeled following the linear Euler-Bernoulli assumptions. The first two eigenvalues of all beams are numerically obtained using Newton-Raphson method. The closed-form expressions of TED and FS of all beams are derived analytically. The influences of Classical dynamical coupled (CL), Lord-Shulman (LS) & Green-Lindsay (GL) thermoelasticity theories, beam dimensions, BaTiO3 volume fraction (Ω f ), and the first two modes (M 1 & M 2) on the TED & FS are meticulously analyzed. Critical thickness (CrTh), critical length (CrLt), and TED (inverse Quality factor) of the beams are numerically obtained and studied. Among other key outcomes, the existence of a critical value of Ω f is established in the range Ω f ∈ [0.5, 0.55], at which, the TED and FS display a drastic change in their natures. The outcomes of the present analysis may find immense potential uses in the design and development of PEMT composite micro-beams, and their applications in several areas such as supporting/stiffening other micro/nanostructures, construction works, sensitive sensing applications, etc.

Analysis of thermoelastic damping and modeling of piezothermoelastic plate resonators with voids

A comprehensive micro-electro mechanical systems (MEMS) model for analysis of a smart composite piezothermoelastic thin plate with voids, in context of Lord-Shulman theory of thermoelasticity has been developed. The work culminates in the derivation of the closed form analytic expressions for thermoelastic damping, frequency shift, electric potential change, volume fraction change, and displacements. Some numerical illustrations and graphical interpretations of analytical developments in case of PZT-5A material piezothermoelastic plate under clamped-simply supported (CS) and simply supported-simply supported (SS) boundary conditions have been presented with the help of MATLAB programming software.

Frequency shifts and thermoelastic damping in distinct Micro-/Nano-scale piezothermoelastic fiber-reinforced composite beams under three heat conduction models

The present work aims to fulfill two primary objectives. Firstly, to present the micro-mechanics model of piezothermoelastic fiber-reinforced composite (PTFRC) medium using analytical techniques, viz., Strength of Materials (SM) & Rule of Mixtures (RM) and graphically establish some of its electro-mechanical-thermal advantages over monolithic PT materials. Secondly, to analytically study the thermoelastic damping (TED) and frequency shift (FS) in micro-/nano-scale PTFRC beams. TED and FS are analyzed considering the linear Euler-Bernoulli theory when the beam is under four distinct boundary conditions, viz., clamped-clamped (CC), simply supported-simply supported (SS), clamped-free (CF), and clamped-simply supported (CS). The eigenvalues of each beam are calculated numerically with 20 iterations each using the Newton-Raphson technique, followed by the analysis of TED and FS considering the assumed boundaries. The Classical dynamical coupled (CL), Lord-Shulman (LS), and Green-Lindsay (GL) thermoelasticity theories are exclusively analyzed. The effects of thermal relaxation parameters, fiber volume fraction, micro and nano beam dimensions, and the first two modes (M1 & M2) on the TED and FS are graphically illustrated. The critical thickness (CrTh) and critical length (CrLt) of all four beams influenced by the existing parameters are scrupulously investigated. The robustness of the obtained results is validated by matching them with previous literature. For high Ωf, the PTFRC beam has superior electro-mechanical-thermal properties and greater Q-factors compared to monolithic beam. However, the Q-factor increases as Ωf decreases. This facilitates the choice of a suitable value of Ωf for designing and optimizing frequency-sensitive microelectromechanical (MEM) and nanoelectromechanical (NEM) PTFRC beams.

Free vibration and buckling characteristics of porous functionally graded materials (FGMs) micro‐beams based on the modified couple stress theory

AbstractBased on the modified couple stress theory (MCST), the free vibration and buckling characteristics of porous functionally graded materials micro‐beams (P‐FGMs and E‐FGMs) are studied. Firstly, two types of pore distribution are considered for functionally graded materials (FGMs) micro‐beams. The governing differential equations of free vibration and buckling of FGMs micro‐beams are derived by Hamilton's principle and dimensionless. Then, the dimensionless governing differential equations and their boundary conditions are transformed by differential transformation method (DTM), then the equivalent algebraic characteristic equations with natural frequencies are obtained. Finally, the dimensionless fundamental frequency and critical buckling load of the free vibration of the porous FGMs micro‐beam under four different boundary conditions of clamped‐clamped (C‐C), clamped‐simply supported (C‐S), simply supported‐simply supported (S‐S) and clamped‐free (C‐F) are calculated. The results are compared with the existing literature data, and the correctness is verified. The effects of porosity, scale parameter, stiffness ratio, axial load and gradient index on the dimensionless natural frequency and critical buckling load of FGMs micro‐beams are discussed.

Approximate closed-form solution for buckling of orthotropic plates with longitudinal edges elastically restrained against rotation

A simple trigonometric series is first introduced to satisfy the longitudinal edges with the clamped-simply supported (CS) condition, while the deformation shape function along the transverse direction is uniquely constructed through a weighted combination of three kinds of trigonometric series to meet the arbitrarily elastically restrained boundary condition against rotation. Approximate closed-form solution for buckling behavior of the orthotropic plates under compression, in-plane shear or combined shear and compression is further presented based on the Galerkin​ method. The transverse edges of the considered orthotropic plates are simply supported, while the opposite longitudinal edges are arbitrarily elastically restrained against rotation to different degree. In particular, the explicit closed-form buckling solutions for the long orthotropic plates under the respective pure in-plane shear and pure compression are obtained. The validity study demonstrates that the relative error of compressive buckling load with a maximum difference of 7% decreases with the increasing of the transverse vs. longitudinal compression load parameter (κ21), while the relative error of critical shear buckling load with a maximum difference of 10% decreases with an increase in the longitudinal compression vs. shear load (κ13) and transverse compression vs. shear load (κ23) parameters. The present approximate closed-form solution is effective and relatively accurate for performing the buckling analysis of orthotropic plates with the longitudinal edges arbitrarily elastically-restrained against rotation, and it can be used in simplified discrete plate analysis of thin-walled composite structures to predict their local buckling strength.

Frequency shifts and thermoelastic damping in different types of Nano-/Micro-scale beams with sandiness and voids under three thermoelasticity theories

The present work focuses on the analytical study of thermoelastic damping (TED) and frequency shift (FS) in micro-scale beams and TED in nano-scale beams. The considered thermoelastic beam is homogeneous, isotropic, and consists of sandiness and voids (TESV). TED and FS of the beam are studied under four types of distinct boundary conditions, viz. clamped-clamped (CC), simply supported-simply supported (SS), Cantilever, and clamped-simply supported (CSS), considering different types of practical engineering applications. Newton–Raphson method is used to numerically evaluate the first two eigenvalues of each type of beam. The expressions of the characteristic roots are also provided. Based on the linear Euler–Bernoulli theory, the closed-form expressions for the beams’ transverse vibrations are derived. Analytical expressions for the deflection, change in temperature, and void volume fraction are also obtained, followed by TED and FS analysis for the considered boundaries. Three theories of thermoelasticity, viz. the Classical dynamical coupled (CL), Lord–Shulman (LS), and Green–Lindsay (GL) thermoelasticity theories are considered in this problem. The influences of thermal relaxation times owing to the three thermoelasticity theories, voids, sandiness, micro & nano dimensions of the beam, and the first two modes (M1 & M2) on the TED and FS are illustrated graphically. Critical thickness (CrTh) and critical length (CrLt) of all four beams under the influence of the existing parameters are meticulously analyzed. Some special cases of this problem are discussed, and the obtained outcomes are validated with well-established results found in the extant literature. In-depth analysis of the effects of thermoelastic coupling and other existing parameters on the natural frequency of micro-/nano-scale resonators, among other applications, is quintessential for designing and optimizing frequency-sensitive nanoelectromechanical systems (NEMS) and microelectromechanical systems (MEMS). Hence, the present analysis may have significant application and contribution in areas utilizing NEMS/MEMS devices.

Effect of thermal axial load on vibration of cracked nanomaterial beam using nonlocal elasticity theory under different boundary conditions

      The aim of this paper is to investigate the free vibration of single-cracked nanomaterials beams under thermal axial load. The beam model is Euler-Bernoulli. The well-known nonlocal elasticity theory is used for analyzing the nanomaterial beams in which the size effect nonlocal parameter is considered. Crack is modeled as a rotational spring that connects the beam segments with each other. The thermal load acts as an axial force on the nanomaterial beam. The effect of the thermal load, the crack location, the crack severity, and the nonlocal parameter are examined in this paper. Two cases of the non-cracked nanomaterial beam and the single-cracked nanomaterial beam are analyzed for three different types of the boundary conditions as simply supported (SS), clamped-clamped (CC), and clamped-simply supported (CS). The results show that when the crack severity is increased the natural frequencies are decreased but in some cases in which the nonlocal parameter value is high, the reverse phenomenon occurs. The temperature changes have a great effect on the frequencies as the temperature is decreased to a value lower than the room temperature, the natural frequencies for all modes decrease and when the temperature is increased to a value higher than the room temperature, the all mode frequencies increase.

On modal analysis of bi-direction FGM beam under general end conditions

This research deals with modal study of bi-direction functionally graded beam. Material characteristics are assumed to be varied continuously through bi-direction i.e. longitudinally and thickness direction according with power law gradation. The finite element modal is created using FE application software COMSOL to analyses the modal characteristics of the bi-direction functionally graded beam. The proposed study is capable of acquiring natural frequencies and mode shapes under general end conditions namely clamped (C-C), simply supported (S-S), clamped-simply supported (C-S) and clamped–free (C-F). The results are validated with the results published in literature for special cases. The effect of aspect ratios is also reported.

Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al2O3) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e0a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory (NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients, which makes it benefit from both hardening and softening effects in small-scale structures. In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported (SS) boundary, the clamped-clamped (CC) boundary, the clamped-free (CF) boundary, and the clamped-simply supported (CS) boundary. Effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. Results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam. The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Semi-Analytical Solution of In-Plane Vibrations of Circular Arches Carrying Added Point Masses

In spite of the current use in modern civil engineering of straight structural components made of reinforced concrete, such as beams and slabs, arches still remain quite often adopted in architectural design, due to their interesting mechanical and esthetical properties. Therefore, practical analytical and numerical tools should be available in order to allow designers to ensure the resistance and stability of this type of structures, especially in the dynamic regime and when point masses are added. The sixth degree differential equation of motion of inextensible arches is known to be quite difficult to deal with and is not applicable to the case examined here of arches with added point masses [1]. The procedure adopted in this article consisted on finding relatively easy particular solutions of this equation satisfying the end conditions and using them as trial functions in the Rayleigh-Ritz formulations of the practical cases of interest. Circular arches of different opening angles are examined here under the classical hypotheses: (1) the effect of shear deformation and rotary inertia are neglected (2) the arch axis is inextensible (3) the dimensions of the cross-section are small in comparisons with the radius;(4) the cross-section is uniform. Three different kinds of end conditions are considered: simply supported-simply supported (SS), clamped-simply supported (CS) and clamped-clamped (CC). The Rayleigh-Ritz method allowed in each case to determine, via an algebraic solution of the associated eigenvalue problem, the first seven mode shapes and natural frequencies of arches with different values of the opening angle. The comparison with the results available in the bibliography was satisfactory and the mode shapes were plotted. The second part of this work was devoted to arches with added concentrated masses: likewise, the numerical values found for the natural frequencies compare quite well with the few references available and the mode shapes are plotted showing the effects of the opening angle, the mass and the positions of the added point masses.

Size-dependent vibration in two-directional functionally graded porous nanobeams under hygro-thermo-mechanical loading

In this study, vibrational behavior of a two-directional functionally graded (2D-FG) porous nanobeam is investigated under hygro-thermo-mechanical loading, based on the Euler-Bernoulli beam theory for various boundary conditions. The properties of the 2D-FG nanobeam are assumed to be in the form of power-law theory. Furthermore, using Eringen’s nonlocal elasticity theory, Hamilton’s principle is utilized to derive the governing equation. Subsequently, the governing equation is converted to an ordinary differential equation using the Galerkin method. The obtained equation is then solved through the Ritz averaging technique for simply-simply supported (S-S), clamped-simply supported (C-S), and clamped-clamped (C-C) boundary conditions to achieve the frequency response of the 2D-FG nanobeam. To validate the obtained equation and the solution procedure, the obtained numerical results are compared with those of a well-known reference in the literature, under similar boundary conditions, material properties, and the values of porosity index and nonlocal parameter, through which the good agreement between the results is demonstrated. Finally, influences of various parameters including moisture (hygro-)effect, porosity, thermal change, nonlocal parameters, and power-law indices in the x - and z -directions are examined for various boundary conditions.

Nonlocal FEM Formulation for Vibration Analysis of Nanowires on Elastic Matrix with Different Materials

In this study, free vibration behaviors of various embedded nanowires made of different materials are investigated by using Eringen’s nonlocal elasticity theory. Silicon carbide nanowire (SiCNW), silver nanowire (AgNW), and gold nanowire (AuNW) are modeled as Euler–Bernoulli nanobeams with various boundary conditions such as simply supported (S-S), clamped simply supported (C-S), clamped–clamped (C-C), and clamped-free (C-F). The interactions between nanowires and medium are simulated by the Winkler elastic foundation model. The Galerkin weighted residual method is applied to the governing equations to gain stiffness and mass matrices. The results are given by tables and graphs. The effects of small-scale parameters, boundary conditions, and foundation parameters on frequencies are examined in detail. In addition, the influence of temperature change on the vibrational responses of the nanowires are also pursued as a case study.

Analytical Solutions for Buckling Behavior of Two Directional Functionally Graded Beams Using a Third Order Shear Deformable Beam Theory

This paper is dedicated to present a Ritz-type analytical solution for buckling behavior of two directional functionally graded beams (2D-FGBs) subjected to various sets of boundary conditions by employing a third order shear deformation theory. The material properties of the beam vary in both axial and thickness directions according to the power-law distribution. The axial, transverse deflections and rotation of the cross sections are expressed in polynomial forms to obtain the buckling load. The auixiliary functions are added to displacement functions to satisfy the boundary conditions. Simply supported – Simply supported (SS), Clamped-Simply supported (CS), Clamped – clamped (CC) and Clamped-free (CF) boundary conditions are considered. Computed results are compared with earlier works for the verification and convergence studies. The effects of the different gradient indexes, various aspect ratios and boundary conditions on the buckling responses of the two directional functionally graded beams are investigated.

Bending analysis of composite and sandwich beams using Ritz method

In the present paper, the bending behaviour of laminated composite and sandwich beams subjected to various sets of boundary conditions which are simply supported (SS), clamped-simply supported (CS), clamped-clamped (CC) and clamped-free (CF) are investigated by using the Timoshenko beam theory and the Ritz method. In order to solve the problem, the shape functions for axial, transverse deflections and the rotation of the cross-section are presented in polynomial forms. The validation and convergence studies are performed by solving symmetric and anti-symmetric cross-ply composite beam problems with various boundary conditions and aspect ratios by adding auxiliary functions to the shape functions. The results in terms of mid-span deflections, axial and shear stresses are compared with those from previous studies to validate the accuracy of the present study. The effects of fiber angle, lay-up and aspect ratio on displacements and stresses are studied.

Comprehensive nonlocal analysis of piezoelectric nanobeams with surface effects in bending, buckling and vibrations under magneto-electro-thermo-mechanical loading

The bending, buckling and vibrational behavior of size-dependent piezoelectric nanobeams under thermo-magneto-mechano-electrical environment are investigated by performing a parametric study, in the presence of surface effects. The Gurtin-Murdoch surface elasticity and Eringen’s nonlocal elasticity theories are applied in the framework of Euler–Bernoulli beam theory to obtain a new non-classical size-dependent beam model for dynamic and static analyses of piezoelectric nanobeams. In order to satisfy the surface equilibrium equations, cubic variation of stress with beam thickness is assumed for the bulk stress component which is neglected in classical beam models. Results are obtained for clamped - simply-supported (C-S) and simply-supported - simply-supported (S-S) boundary conditions using a proposed analytical solution method. Numerical examples are presented to demonstrate the effects of length, surface effects, nonlocal parameter and environmental changes (temperature, magnetic field and external voltage) on deflection, critical buckling load and natural frequency for each boundary condition. Results of this study can serve as benchmarks for the design and analysis of nanostructures of magneto-electro-thermo-elastic materials.

Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory

This paper presents the free vibration behavior of two directional functionally graded beams subjected to various sets of boundary conditions which are simply supported (SS), clamped-simply supported (CS), clamped-clamped (CC) and clamped-free (CF) by employing a third order shear deformation theory. The material properties of the beam vary exponentially in both directions. In order to investigate the free vibration response, the equations of motion are derived by means of Lagrange equations. The axial, transverse deflections and rotation of the cross sections are expressed in polynomial forms including auxiliary functions which are used to satisfy the boundary conditions. The verification and convergence studies are performed by using computed results from a previous study which is based on the Timoshenko beam formulation. The results for extensive studies are provided to understand the influences of the different gradient indexes, various aspect ratios and boundary conditions on the free vibration responses of the two directional functionally graded beams.

Static Analysis of Reddy-Bickford Composite and Sandwich Beams via Ritz Method

This paper is dedicated to present static behaviour of Reddy-Bickford laminated composite and sandwich beams subjected to various sets of boundary conditions which are simply supported (SS), clamped-simply supported (CS), clamped-clamped (CC) and clamped-free (CF) by using Ritz method. An analytical solution based on polynomial series including auixiliary functions which are used to satisfy the boundary conditions is developed to solve the studied problem. The polynomial shape functions for axial, transverse deflections and the rotation of the cross-section are presented. The validation and convergence studies are performed by solving symmetric and anti-symmetric cross-ply composite beam problems with various boundary conditions and aspect ratios. The numerical results in terms of mid-span deflections, axial and shear stresses are obtained to make comparison with previous studies and to investigate the accuracy of the present study. The effects of fiber angle, lay-up and aspect ratio on displacements and stresses are studied. The static response of the various laminated composite sandwich structures which have symmetric lay-up based on the various boundary conditions, fiber angles and thickness ratios is also studied. It is found that the polynomial series with auxiliary functions can be used for the static analysis of the composite and sandwich beams via Ritz method.

Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory

In this study, free vibration behavior of piezoelectric Timoshenko nanobeams in the vicinity of postbuckling domain is investigated based on the nonlocal elasticity theory. It is assumed that the piezoelectric nanobeam is subjected to an axial compression force, an applied voltage and a uniform temperature change. Using Hamilton principle, the governing differential equations of motion incorporating von Kármán geometric nonlinearity and the corresponding boundary conditions are derived and then discretized on the basis of generalized differential quadrature (GDQ) scheme. After solving the parameterized equations using Newton–Raphson technique, a dynamic analysis based on a numerical solution strategy is performed to predict the natural frequencies of piezoelectric nanobeams associated with both prebuckling and postbuckling domains. Numerical results are presented to study the effects of nonlocal parameter, temperature rise and external electric voltage on the size-dependent vibration behavior of piezoelectric nanobeams with clamped–clamped (C–C), clamped-simply supported (C-SS) and simply supported-simply supported (SS-SS) end conditions. It is demonstrated that these parameters may shift the postbuckling domain to higher or lower applied axial loads.