Third-order Shear Deformation Beam Theory Research Articles

Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams.

This paper examines the accuracy and effectiveness of various beam theories in predicting the critical buckling loads and fundamental frequencies of functionally graded porous (FGP) beams whose material properties change continuously across the thickness. The beam theories considered are classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and the broken-line hypothesis-based shear deformation beam theory (BSDBT). Governing equations for those beam theories are formulated by using the Hamilton's principle and are then solved by means of the generalised differential quadrature method. Finite element simulation solutions are provided as reference results to assess the predictions of those beam theories. Comprehensive numerical results are presented to evaluate the influences of the porosity distribution and coefficient, slenderness ratio, and boundary condition on the difference between theoretical predictions and simulation results. It is found that the differences significantly increase as the porosity coefficient rises, and this effect becomes more noticeable for the rigid beam with a smaller slenderness ratio. Nonetheless, the results produced by the BSDBT are always the closest to simulation ones. The findings in this paper will contribute to the establishment of more refined theories for the mechanical analysis of FGP structures.

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.

Statistical analysis and Neural Network Modeling of functionally graded porous nanobeams vibration in an elastic medium by considering the surface effects

The natural frequency of a clamped–clamped functionally graded porous (FGP) nanobeam is predicted in this study. Material distribution is considered based on monotonous, symmetric, and non-symmetric patterns in the thickness direction. This paper deals with governing equations of nanobeams based on third-order shear deformation beam theory in conjunction with nonlocal strain gradient theory (NSGT) and surface effects. Artificial neural network (ANN) is utilized to predict the effect of eight parameters including temperature gradient, residual surface stress, porosity distribution pattern, porosity parameter, nonlocal and material length scale parameters, and elastic and shear coefficients of Pasternak foundation on the fundamental frequency of FGP nanobeam. Different training methods are selected to simulate input and output dependency. Results show that the dependency of the natural frequency is inverse to the temperature gradient and nonlocal parameter in the sense that increasing these factors will decrease the natural frequency. Also, increasing the material length scale parameter grows the effect of the nonlocal parameter. Residual surface stress, material length scale, and Pasternak foundation parameters have a direct effect on the output and among them; the material length scale parameter has a more noticeable effect. Finally, it was found that by increasing the porosity parameter value, the diversity of natural frequency levels up drastically

Supersonic flutter and free vibration features of functionally graded material nanobeams incorporating surface effects

In the present work, a numerical investigation is provided for free vibrational and aeroelastic stability features of nanobeams made of Functionally Graded Material (FGM) resting on an elastic Pasternak foundation. The FGM nanobeam's mechanical properties are considered to differ based on power-law and exponential functions in the thickness direction. The effect of size is modeled in terms of the Gurtin-Murdoch surface elasticity theory and the Nonlocal Strain Gradient Theory (NSGT) is employed for incorporating the surface effects. Mathematical modeling of the FGM nanobeam is oriented by three various beam theories including Reddy's third-order Shear Deformation Beam Theory (TSDBT), Timoshenko beam theory (TBT), and Euler-Bernoulli Beam Theory (EBT), moreover, the aerodynamic pressure is determined in terms of the linear supersonic piston theory. It is concluded that when the surface effects are incorporated, the width-to-length ratio of the FGM nanobeam plays a sensible role in determining the natural frequencies; but it has no sensible effect on the flutter boundaries. Numerical results reveal that the importance of the surface effects increases as the values of thickness-to-length and width-to-length ratios decrease.

Thermal Backbone Curves of Nanocomposite Beams Reinforced with Graphene Platelet on Elastic Foundation

In this paper, an examination on the backbone curves of nanocomposite beams reinforced with graphene platelet (GPL) on elastic foundation exposed to a temperature increment is accomplished. By means of the Hamilton principle and in the framework of a third-order shear deformation beam theory, i.e. Reddy’s beam theory (RBT), along with the von Karman nonlinear strains, the nonlinear motion equations are disclosed. The Halpin–Tsai micromechanical model is exploited in order to release the effective modulus of elasticity of the nanocomposite beam while the thermal expansion coefficient, Poisson’s ratio and the mass density are estimated resorting to the rule of mixtures. Elastic foundation, and uniform temperature (UD) rising impacts are incorporated during the mechanical modeling of the nanocomposite beam. The Ritz scheme together with an iterative procedure reveals the nonlinear natural frequency associated to an assumed deflection in order to sketch the corresponding backbone curve. The outcomes are validated in comparison with the available results. Some case studies are established for the sake of clarifying the impressions of the distribution pattern of the GPL, the beam length to its thickness ratio, the weight fraction of the GPL, the elastic foundation, the boundary condition type, and the temperature changes on the first backbone curve of the nanocomposite beam. It is elucidated that the increment of the weight fraction of the GPL with X/O distribution pattern decreases/increases the hardening behavior of clamped–clamped (C-C) and simply-supported beam, while the softening behavior of a clamped-free (C-F) nanocomposite beam is independent of the division pattern, and the weight fraction of the GPL. Moreover, the temperature increment unlike the elastic foundation develops the hardening behavior of the backbone curves of simply supported and C-C nanocomposite beams. Although the backbone curve associated to a C-F nanocomposite beam is invariant with respect to the temperature, the elastic foundation develops the softening trend of C-F nanocomposite beams.

Influence of Graded Surface Decarburization of Automobile Forging Front Axle on the Bending Behavior Based on a Third-Order Shear Deformation Beam Theory

During the forging process of automobile front axle, the steel near the surface is often decarburized for a certain depth. The mechanical properties at the decarburization layer are graded and different from the inner area, influencing the bending behavior of axles under heavy loads. In this paper, the decarburized forging of front axle is regarded as a rectangular thick sandwich beam, composed of a homogeneous core and the functionally graded layer coated on both bottom and top surface. A Third-order Shear Deformation Theory (TSDT) is employed to investigate the static bending behaviors under two point−loads. The properties of sandwich FG material are represented with a piecewise power−law function, and the displacement field governing equations are derived through the virtual work principle. The Navier analytical method and numerical DQM procedures are employed to obtain the bending responses under simply supported boundary conditions, and the results are validated through the comparison with an example in the literature. Then, the transverse deflection, rotation, axial stress, and shear stress are studied in terms of different power−law exponents, decarburization depth, unsymmetrical decarburization depth, unbalance loading, and beam sectional dimension. The study reveals the influence of surface decarburization on the bending behavior of forged automobile front axles, and contributes to the optimization of structure design.

Dynamic instability of nanocomposite piezoelectric‐leptadenia pyrotechnica rheological elastomer‐porous functionally graded materials micro viscoelastic beams at various strain gradient higher‐order theories

AbstractThe dynamic stability response of a micro sandwich beam with leptadenia pyrotechnica rheological elastomer (LPRE) core is studied. The top and bottom layers respectively are assumed as piezoelectric reinforced with carbon nanotubes (CNTs) and porous functionally graded materials (FGM). The core and top layers are affected by magnetic and electric fields for the magnetic and piezoelectric characteristics of the layers, respectively. The Halpin‐Tsai micromechanics theory for obtaining the effective material properties of the nanocomposite layer is utilized. On the basis of Kelvin‐Voigt model, the structural damping of the smart micro beam is assumed. The microstructure is located on the viscoelastic model which is simulated using Visco‐Pasternak platform. The size effects are assumed according to the theory of strain gradient including three‐length scale constants. The various theories of first‐order shear deformation beam theory (FSDBT), third‐order shear deformation beam theory (TSDBT), parabolic shear deformation beam theory (PSDBT), and exponential shear deformation beam theory (ESDBT) are utilized for driving the governing equations according to the Hamilton's principle. The motion final relations are solved by the differential quadrature method (DQM) for presenting the dynamic buckling area. The effects of different components such as volume percentage and distribution of GPLs, porosities, magnetic field of LPRE, applied voltage, FG index, structural damping, and geometric components of the micro sandwich beam on the dynamic stability reign (DIR) of the system are shown. The results with other researcher papers are compared. The results show that the DIR increases by applying a magnetic field to the LPRE layer.

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

In present paper, the piezoelectric effect on bifurcation buckling of symmetric FGM porous nanobeam is presented based on a higher-order nonlocal elasticity and strain gradient theory in conjunction with Reddy third-order shear deformation beam theory. The strain energy density and electric enthalpy density functions are used to derive constitutive relations for porous functionally graded, as well as piezoelectric materials. Equations of motion for elastically supported layered FGM nanobeam with diverse distribution of porosity and electro-elastic coupling are derived based on the modified Hamilton’s variational principle. The formulated boundary value problem is analytically solved. For the first time, the effects of porosity distribution, volume of voids, distance between porosity and FGM core surfaces as well as material gradation, layers size, aspect ratio, stiffness of Kerr foundation, and external electric voltage on critical in-plane force, critical porosity and critical voltage were comprehensively presented and discussed. The investigation enables analysis and precise multi-dimensional control of static response of smart beam-like nanostructures widely used in MEMS and NEMS devices.

Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid-plane Formulations

The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surface-based formulation, the governing equations of motion are established. For the case of beams with immovable edges in in-plane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters.

Vibration suppression of magnetostrictive laminated beams resting on viscoelastic foundation

The vibration suppression analysis of a simply-supported laminated composite beam with magnetostrictive layers resting on visco-Pasternak’s foundation is presented. The constant gain distributed controller of the velocity feedback is utilized for the purpose of vibration damping. The formulation of displacement field is proposed according to Euler-Bernoulli’s classical beam theory (ECBT), Timoshenko’s first-order beam theory (TFBT), Reddy’s third-order shear deformation beam theory, and the simple sinusoidal shear deformation beam theory. Hamilton’s principle is utilized to give the equations of motion and then to describe the vibration of the current beam. Based on Navier’s approach, the solution of the dynamic system is obtained. The effects of the material properties, the modes, the thickness ratios, the lamination schemes, the magnitudes of the feedback coefficient, the position of magnetostrictive layers at the structure, and the foundation modules are extensively studied and discussed.

Gram-Schmidt-Ritz method for dynamic response of FG-GPLRC beams under multiple moving loads

The objective of this study is to apply Gram-Schmidt orthogonalization procedure for generating displacement functions. This procedure allows us to obtain numerically stable functions to be used in vibration analysis of novel composite beams via the Ritz method. Within this study, the theoretical modeling based on third-order shear deformation beam theory is derived to deal with free and forced vibration of functionally graded graphene nanoplatelet-reinforced composite beams subjected to multiple moving loads. To solve the problems of the beams with various end supports, the Ritz method-based Gram-Schmidt polynomial series is adopted to obtain the accurate results. In forced vibration due to the applied multiple moving loads, the time integration technique of Newmark is also employed to find out time history of dynamic deflection of such beams associated with several parametric studies such as patterns of reinforcement, weight fraction of graphene nanoplatelets, and the number of moving loads. According to the numerical results, it is found that the composite beams with low content of graphene nanoplatelets can result in enhancing performance of dynamic bending resistance. Increasing number of moving loads on the beams yields the increases of dynamic deflection and critical velocity.

A refined beam theory for bending and vibration of functionally graded tube-beams

This paper presents a novel approach for analyzing transverse bending and vibration of functionally graded circular cylindrical tubes with radial nonhomogeneity. Different from the Euler-Bernoulli and Timoshenko theories of beams, a refined beam theory or third-order shear deformation beam theory for radially graded tubes is proposed, where warping, shear deformation and rotational moment of inertia of cross-section are all considered. The shear correction coefficient is not needed. Coupled governing equations for the deflection and rotation about the neutral axis of cross-section are derived from equilibrium equations, and then converted to a single fourth-order partial differential governing equation. The deflection and stress distribution for cantilever and simply-supported tubes are derived explicitly. The frequency equations for free flexural vibration of radially graded hollow cylinders with clamped-clamped, pinned-pinned, and clamped-free ends are obtained and the natural frequencies are calculated for different power-law gradients and various length/thicknesses ratios. The effects of radial gradient on the stress distribution and the natural frequencies are analyzed in detail.

Nonlinear Dynamics of Temperature-Dependent FG-GRC Laminated Beams Resting on Visco-Pasternak Foundations

This paper studies the nonlinear dynamic responses of graphene-reinforced composite (GRC) beams in a thermal environment. It is assumed that a laminated beam rests on a Pasternak foundation with viscosity and consists of GRC layers with various volume fractions of graphene reinforcement to construct a functionally graded (FG) pattern along the transverse direction of the beam. An extended Halpin–Tsai model which is calibrated against the results from molecular dynamics (MD) simulations is used to evaluate the material properties of GRC layers. The mechanical model of the beam is on the establishment of a third-order shear deformation beam theory and includes the von-Kármán nonlinearity effect. The model also considers the foundation support and the temperature variation. The two-step perturbation technique is first applied to solve the beam motion equations and to derive the nonlinear dynamic load–deflection equation of the beam. Then a Runge–Kutta numerical method is applied and the solutions for this nonlinear equation are obtained. The influence of FG patterns, visco-elastic foundation, ambient temperature and applied load on transient response behaviors of simply supported FG-GRC laminated beams is revealed and examined in detail.

Dynamic response of a size-dependent nanobeam to low velocity impact by a nanoparticle with considering atomic interaction forces

In the present paper, low velocity impact response of a size-dependent nanobeam in a thermal field with uniform temperature distribution has been investigated. The van-der Waals interaction force based on description of Lennard–Jonses is considered as the impact force between nanoparticle and nanobeam. According to third-order shear deformation beam theory, the governing equations are obtained using Hamilton's principle based on nonlocal strain-gradient theory. The Galerkin's method was adopted to solve the differential equations of nanobeam with simply supported and clamped boundary conditions. Afterward, the system of time-dependent equations by applying the fourth-order Runge–Kutta method is solved. The parametric study is presented to examine the effect of particle radius, initial velocity, temperature environment, the nonlocal parameter, and the length-scale parameter on the impact response of nanobeam.

Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams

In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.

Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams

In this paper, the static and dynamic responses of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties vary along both thickness and axial directions. Employing Hamilton’s principle, the differential equations are derived based on von-Karman geometric nonlinearity and third-order shear deformation beam theory. The modified couple stress theory is adopted to capture the size effects. The material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The differential equations and boundary conditions are discretized using differential quadrature method. Subsequently, static bifurcation of microbeams is calculated utilizing pseudo-arc length continuation technique. The free vibration around the postbuckling configuration of microbeams is studied by solving the associated linear eigenvalue problem. The influences of the functional gradient parameters, dimensionless length scale parameter and aspect ratio on the static response and vibration characteristics of buckled BDFG microbeams with various boundary conditions are investigated. It is demonstrated that the buckling of BDFG microbeam occurs through a transcritical bifurcation. The static stable responses of BDFG microbeam are asymmetric due to the stretching-bending coupling that results from the asymmetry of the material distribution in thickness direction of beam. The mode veering phenomenon is detected in postbuckling domain of microbeams. It is examined that the longitudinal dynamic displacement plays a pivotal role in the mode veering phenomenon.

Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium

• Material properties of microbeam vary along both thickness and axial directions . • Material length scale parameter is a function of spatial coordinates and varies with the gradient parameters. • It shows that the variable scale parameter has significant influence on the static and dynamic characters of microbeam. In this paper, size dependent free vibration , buckling and dynamic stability of bi-directional functionally graded (BDFG) microbeam embedded in elastic medium are investigated. The material properties vary along both thickness and axial directions. In particular, the material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The system of differential equations with variable coefficients governing the motion of BDFG microbeam is derived employing Hamilton’s principle, the modified couple stress theory and third-order shear deformation beam theory. The differential quadrature method (DQM) is utilized to solve the static and dynamic problem. Three different models evaluating the material length scale parameter of BDFG microbeam are presented for comparison. Parametric studies are carried out to show the influence of gradient parameters, size effect, stiffness of elastic medium on the free vibration, buckling and dynamic stability characteristic of BDFG microbeam. Results show that the variation of material length scale parameter should be considered in the analysis of BDFG microbeam.

Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams

An inhomogeneous beam model incorporating size effects is formulated based on the nonlocal strain gradient theory of elasticity within the framework of a third-order shear deformation beam theory to analyze the size dependency in vibration behavior of postbuckled laminated functionally graded (FG) micro-/nanobeams made from graphene platelet-reinforced composite (GPLRC). The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the nanofiller varies layerwise on the basis of different patterns of FG dispersion. Based upon the Halpin–Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC micro-/nanobeams are extracted. With the aid of the Hamilton’s principle, the nonlocal strain gradient equations of motion are constructed. After that, an improved perturbation technique is put to use to capture the small-scale effects on the fundamental frequencies of laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. The size-dependent natural frequencies of laminated third-order shear deformable FG-GPLRC micro-/nanobeams are obtained as function of applied axial compressive load within both the prebuckling and postbuckling domains. It is seen that in the prebuckling regime, the softening-stiffness effect of the nonlocality causes to reduce the natural frequencies of laminated FG-GPLRC micro-/nanobeam, but the hardening-stiffness influence of strain gradient size effect leads to increase the natural frequencies. However, by moving to the postbuckling regime, this pattern becomes vice versa, as the nonlocal size dependency leads to increase the natural frequencies of micro-/nanobeam while the strain gradient size effect causes to reduce them.

Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams

• Nonlinear behaviour is of a hardening type. • Both symmetric and asymmetric modes are present. • Larger gradient index results in shift in resonance. This paper aims at investigating the nonlinear vibration characteristics of axially functionally graded (AFG) shear deformable tapered beams subjected to external harmonic excitations. A coupled nonlinear longitudinal-transverse-rotational set of equations governing the motion of the AFG system is derived utilising the third-order shear deformation beam theory via Hamilton's energy principle. The beam under consideration is tapered; i.e. the width of the beam varies along the length. The tapered geometry along with the nonuniform material properties arising from the AFG nature of the beam increases the complexity in the modelling and numerical simulations. The expressions for the kinetic and potential energies of the AFG shear-deformable tapered beam together with the expressions for the work of damping and external excitation are derived and implemented in Hamilton's principle. The nonlinear partial differential equations are discretised making use of the Galerkin technique and solved with the aid of a continuation scheme. The effect of different parameters such as the gradient index and the tapered ratios on the force- and frequency-amplitude diagrams of the AFG system is examined.

Geometrically nonlinear analysis of sandwich beams under low velocity impact: analytical and experimental investigation

Nonlinear low velocity impact response of sandwich beam with laminated composite face sheets and soft core is studied based on Extended High Order Sandwich Panel Theory (EHSAPT). The face sheets follow the Third order shear deformation beam theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the two dimensional elasticity is used for the core. The nonlinear Von Karman type relations for strains of face sheets and the core are adopted. Contact force between the impactor and the beam is obtained using the modified Hertz law. The field equations are derived via the Ritz based applied to the total energy of the system. The solution is obtained in the time domain by implementing the well-known Runge-Kutta method. The effects of boundary conditions, core-to-face sheet thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that each of these parameters have significant effect on the impact characteristics which should be considered. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The contact force histories predicted by EHSAPT are in good agreement with that obtained by experimental results.