Higher-order Shear Deformation Beam Research Articles

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

An enhanced quasi-3D HSDT for free vibration analysis of porous FG-CNT beams on a new concept of orthotropic VE-foundations

The original primary objective of this study is to conduct a comprehensive investigation into the free vibration behavior of beams with porosity, reinforced by carbon nanotubes (CNTs). These beams are supported by an arbitrary orthotropic variable elastic foundation (AOVEF). The focus lies on understanding how the presence of porosity and CNT reinforcement, coupled with the complex support provided by the AOVEF, influences the vibration characteristics of the beams. In addition, we intend to examine the impact of a number of micromechanical models on the vibration properties of these beams. Four carbon nanotube variables are introduced to symbolize several CNT variations. CNTs’ mechanical properties are examined through a variety of micromechanical models. Shear deformation effects are considered into account in the framework of higher-order shear deformation (HSDT) beam theory. The equations of motion are constructed using Hamilton’s concept and the equation system for the FG-CNT beam with supported ends is solved via the Navier methodology. A new approach in elastic foundation modeling is the Arbitrarily Orthotropic Variable Elastic (AOP-VE) foundation. It builds on the Winkler layer concept but introduces the idea of a variable response along the length of the beam foundation. Unlike the Winkler-Pasternak model, AOP-VE allows for control over the directional properties of the Pasternak foundation. The study looks into multiple aspects, involving various distribution kinds of CNTs, the impact of Winkler and Pasternak factors, mode values, side-to-length ratio, porosity, and angle variation. The results indicate that these parameters have a major effect on the inherent vibration features of FG-CNT beams. The study results in various novel findings that improve the understanding of the subject topic and set a standard for future studies.

Investigation on internal resonance of fluid conveying pipes with initial geometric imperfection

This paper shows numerical research on the 1:2 internal resonance and chaotic bifurcation phenomena of the graphene platelets reinforced metal foams (GPLRMF) pipes, with emphasis on the effect of initial geometric imperfection. To this end, combining the high order shear deformation beam theory and Kelvin-Voigt viscoelasticity, a mathematical model that considers geometric imperfection is established. The fluid in pipeline is supposed to be incompressible and inviscid. Subsequently, the Galerkin discretization and multiscale method are utilized for solving the nonlinear governing equations in the case of 1:2 internal resonance. And the nonlinear dynamic behavior of the fluid conveying pipes can be analyzed through the frequency amplitude diagrams, bifurcation curves, time course curves, phase trajectories, and Poincaré maps. The double jump phenomena are excited to demonstrate the dynamic characteristics of the pipes with 1:2 internal resonance. In the case of small disturbance coefficients, various internal resonance and chaos phenomena can be captured in the pipeline system. Research shows that the coupled effects between initial geometric imperfection and nonlinear geometric relationships is non-monotonic. That is to say, for bimodal internal resonance, when the initial geometric imperfection increase, the unstable region will be compressed, and the response amplitude will increase.

Geometrically Nonlinear Buckling of FG-CNTRC Plates Stiffened by FG-CNTRC Stiffeners Subjected to Combined Loads Using Nonlinear Reddy’s HSDT

The nonlinear buckling and postbuckling of functionally graded carbon nanotube reinforcement composite (FG-CNTRC) plates resting on the nonlinear effect of elastic foundation under combined load including external pressure and axial compression loads with uniformly distributed temperature rises are fully investigated in this paper. The FG-CNTRC plates are reinforced by FG-CNTRC stiffeners and the plate-foundation interaction is modeled using a nonlinear model. The higher-order shear deformation plate theory with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Additionally, by using the anisotropic higher-order shear deformation beam theory, a new smeared stiffener technique is successfully developed for FG-CNTRC stiffeners. Galerkin’s method is used to achieve the equilibrium equation system in the nonlinear algebraic forms, and the critical load and postbuckling load–deflection curve expression are explicitly determined. The numerical investigations discuss the influences of FG-CNTRC stiffeners, hardening/softening nonlinear elastic foundation, uniformly distributed temperature rises, geometrical and material features on nonlinear stability of stiffened FG-CNTRC plates.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

Porosities-dependent wave propagation in bi-directional functionally graded cantilever beam with higher-order shear model

The contribution provided in this paper is to investigate the dynamic response of bidirectional graded material beams with porosities. Higher order shear deformation beam theory for wave propagation in porous functionally graded cantilevered beam are developed, taking into account the bidirectional distribution, which is mainly represented in the density and the Young’s modulus. The material properties of the porous FG beam are graded through the thickness and width using a power law distribution. The governing equations of the wave propagation in the porous FG beam are derived by employing the Hamilton’s principle. The analytic dispersion relation of the bidirectional porous FG beam is obtained by solving an eigenvalue problem. Two porosities models approximating the even and uneven distributions of porosity are considered. The results obtained for bidirectional FG beam were compared with those reported in the literature, and the results were very agreement. The effects of the volume fraction distributions, the number of wave propagation, various parameters and the porosity type on dynamic of functionally graded beam are discussed in details. The numerical results show that the power law index, number of wave, wave velocity, and porosity distribution model affect the dynamic behavior of the beam significantly. The results show that the uneven porosity predicts higher natural frequencies than the even porosity.

Free vibration and buckling of functional graded box beams and the effect of the cross-sectional warping shape

Box beams have been used widely in engineering fields. This paper proposes a higher-order shear deformation beam theory to investigate the transverse vibration and stability of box beams in pure bending, especially free vibration and buckling of functionally graded box beams. A warping shape function is constructed for the first time to take shear deformation and rotary inertia of the hollow-core rectangular cross section into account simultaneously during bending. The governing equation and boundary conditions are derived. The frequency equations for typical end supports are given. The characteristic equations for buckling and critical loads are determined. Numerical results are evaluated. Besides its simplicity and easy-to-implement, the proposed theory has sufficient accuracy on the mechanical behavior of the box beams. By comparing the obtained results with the simulation results based on three-dimensional finite element method, the accuracy of the numerical results obtained is verified. The effects of parameters such as aspect ratio, gradient index, and wall thickness on the natural frequencies and critical loads are explored in detail. When box beams become rectangular beams, our model reduces to the well-known Levinson beam model.

Re-examination of nonlinear vibration, nonlinear bending and thermal postbuckling of porous sandwich beams reinforced by graphene platelets

This paper re-examines the nonlinear free vibration, nonlinear bending and thermal postbuckling behaviors of porous sandwich beams with graphene platelets (GPL) reinforcements rested on elastic foundations. We remove the equivalent isotropic model (EIM) and introduce an inhomogeneous model. The shear modulus together with the Young’s moduli for porous GPLRC core is determined by a generic Halpin–Tsai model with porosity. Mechanical properties of both porous GPLRC core and metal face sheets are temperature dependent. Based on the framework of higher order shear deformation beam theory, the motion equations of porous sandwich beams with GPL reinforcements are established. In the modeling, the von Kármán kinematic nonlinearity, beam-foundation interaction and thermal effect are considered. By employing the two-step perturbation method, the analytical solutions are obtained for the nonlinear vibration and nonlinear bending as well as thermal postbuckling problems. Numerical investigations are performed for comparing the results obtained from the EIM and the present model. The outcomes reveal that the EIM is invalid for linear free vibration, thermal buckling and postbuckling analyses of porous sandwich beams with GPL reinforcements, but the EIM is valid in some cases for nonlinear bending and nonlinear vibration analyses of the same beam.

Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams

ABSTRACT This paper presents the hygro-thermal vibration analysis of bidirectional functionally graded (BDFG) porous nanobeams resting on elastic foundation based on a nonlocal strain gradient theory and the finite element method. The effects of temperature and moisture on structures are assumed to cause tension load in the plane and do not change the material's mechanical properties. Nanobeams are made from functionally graded (FG) materials which the mechanical properties change along with the thickness (z-axis) and length (x-axis) of nanobeams according to the power-law model. The porosity distribution is considered to be even and uneven. For the first time, a two-node beam element with nine degrees of freedom at each node is developed to analyze the hygro-thermal vibration behavior in the nanoscale. The elastic foundation is used in this work to be the Winkler-Pasternak type. Applying Hamilton's principle based on the refined higher-order shear deformation beam theory (RBT) and the~nonlocal strain gradient model, governing equations of nanobeams are derived. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. The nonlocal coefficient makes the nanobeam softer, while the strain gradient coefficient makes it stiffer. In addition, the effects of the nonlocal coefficient, porosity coefficient, foundation stiffness, boundary conditions on the natural frequency of nanobeams are investigated in detail.

Hygro-thermo-mechanical vibration analysis of functionally graded porous curved nanobeams resting on elastic foundations

This paper presents an analytical solution using Chebyshev polynomials based on Rayleigh–Ritz method and Navier's solution to analyze the free vibration response of functionally graded porous (FGP) curved nanobeams embedded resting on an elastic medium under hygro-thermo-mechanical load. Applying Hamilton's principle based on the quasi-3D higher-order shear deformation beam theory and the nonlocal elasticity theory, the governing equation of FGP curved nanobeam is derived. Material properties of nanobeam continuously change through the thickness via a power-law distribution and porosity distribution is described by two laws including even porosity distribution and uneven porosity distribution, respectively. The impact of thermal and moisture on structures are assumed to cause tension load in the plane and do not change the material properties. The elastic foundation used in this work is the Winkler–Pasternak type. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. In addition, the influence of curvature radius, nonlocal coefficient, porosity coefficient, stiffness of foundation, and boundary conditions on the free vibration of the nanobeam are examined in detail.

Wave propagation in a porous functionally graded curved viscoelastic nano-size beam

In this paper, wave propagation analysis of a viscoelastic system of curved nanobeams made of porous functionally graded materials (P-FGM) is studied, for the first time. The displacement fields for the curved nanobeam are obtained via a higher-order shear deformation beam theory which contains curvilinear axial, rotation, transverse, and thickness stretching effects, while size effects are considered using nonlocal strain gradient theory. The effective material properties for P-FGMs are expressed by modified model of power-law variation which is connected with cosine functions to estimate the effect of porosity and Kelvin–Voigt viscoelastic model for estimating viscoelastic behavior on mechanics of nanostructures. The Hamilton’s principle is applied to find governing equations of motion of the viscoelastic curved nanobeam. The state space method and a set of mathematical series are developed to find response. The Influence of porosity value and types of distributions, viscosity coefficient, power-law index, geometry, and opening angle (related to curvature radius) on phase velocity of curved viscoelastic nanobeams is investigated.

Buckling Analysis of Functionally Graded Beams Using the Finite Element Method

This study developed a finite element model according to higher-order shear deformation beam theory (HSDT) for the buckling analysis of functionally graded (FG) beams. Equilibrium equations of the FG beam are obtained from Lagrange’s equations. The beam element to be discussed within the scope of the study has 5 nodes and 16 degrees of freedom (DOF). As a result of the buckling analysis, the critical buckling load of the beam was obtained for various boundary conditions, power-law index (p), and slenderness (L/h). When the critical buckling loads obtained as a result of the analysis were compared with the literature, it was seen that they were quite compatible.

Magneto-hygro-thermal vibration analysis of the viscoelastic nanobeams reinforcedwith carbon nanotubes resting on Kerr’s elastic foundation based on NSGT

This paper studies the vibration analysis of the viscoelastic nanobeams reinforced with functionally graded carbon nanotubes (FG-CNTs), considering the thickness stretching effect subjected to magneto-hygro-thermo loading based on a novel higher-order shear deformation theory. It is assumed that the nanosize structure is resting on Kerr’s elastic foundation. The effective material properties of FG-CNTs are changed through the thickness direction based on five diverse distributions of CNTs. All of the effective material properties of the nanobeam are temperature-dependent. Nonlocal strain gradient theory is applied to consider the size effect. A higher-order shear deformation beam theory is utilized to consider the shear impacts. Three kinds of relations are employed to study the effects of the hygro-thermo loading comprehensively on the frequency of the nanobeams. An analytical solution technique is utilized to solve the nonlocal governing equations achieved from Hamilton’s principle for several boundary conditions. The influences of size effect parameters, elastic foundation factors, distributions of CNTs, hygrothermal environment, and viscoelastic parameter on the frequency of nanobeams are investigated.

Vibration analysis of laminated composite higher order beams under varying axial loads

The article develops a computational mathematical model to predict the dynamic response of composite laminated beam under variable axial load using a third higher order shear deformation beam theory, for the first time. The geometrical kinematic relations of displacements are portrayed with higher parabolic shear deformation beam theory. Constitutive equations of composite beam are used on the basis of plane stress problem. The variable axial load is distributed along the axial direction by: constant, linear, and parabolic functions. The equations of motion and associated boundary conditions are derived using Hamilton's principle. Then, the variable coefficients-differential equations of motion are discretized in spatial direction by using numerical differential quadrature method (DQM). The proposed model is verified with previous works available in literature. Parametric analyses are developed to present the influence of axial load type, orthotropy ratio, slenderness ratio, fiber orientations, and boundary conditions on the natural frequencies of composite beams. The present enhanced model can be used specially in designing of spacecrafts, naval, automotive, helicopter, the wind turbine, musical instruments, and civil structures subjected to the variable axial loads.

Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory

A finite element modeling combined with the strain gradient theory (SGT) and the refined higher-order shear deformation beam theory is developed to study the dynamic instability of magnetically embedded functionally graded porous (FGP) nanobeams. Nanobeams with elastic foundation (EF) subjected to an axially oscillating load are analyzed. Nanobeams are made of functionally graded material (FGM) with an uneven porosity distribution. A three-node beam element with 8 degrees of freedom (DOFs) for two outer nodes and 2 DOFs for the middle node, which has the C1 and C2 continuous Hermite shape functions, is used to simulate nanobeams. Besides, Bolotin’s method is employed to determine the instability region of FGP nanobeams. The accuracy of the proposed method is tested by comparing it with other published works. In addition, the influences of various parameters such as magnetic potential, small-scale parameter, porosity coefficient, stiffness foundation, boundary conditions (BCs) on the dynamic instability of nanobeams are studied in detail.

Nonlinear Thermo-Mechanical Response of Bi-Directional Functionally Graded Porous Beams with Initial Geometrical Imperfection

In this paper, the nonlinear vibration and buckling of bi-directional functionally graded (2D-FG) beam subjected to thermal loading are examined via the higher-order shear deformable beam theory incorporating the von Kármán geometric nonlinearity. Two types of initial defects including the internal porosity distribution and external geometrical imperfection shape are taken into consideration. The temperature-dependent material property varying along the length and thickness following the power-law function is employed, and the porosities inside the 2D-FG beam with even and uneven distributions are considered. The initial geometrical imperfection is described by the product of trigonometric and hyperbolic functions, which can capture both global and localized imperfection shapes. The discrete equations of motion for the 2D-FG beam are established by using the differential quadrature method (DQM), and the derived function is solved by an iterative procedure. The nonlinear thermo-mechanical vibration and buckling of 2D-FG beams under uniform, linear, and nonlinear temperature loads are systemically compared. Moreover, several key factors such as 2D-FG indexes, porosity distribution, geometrical imperfections, as well as boundary conditions are investigated in detail. The proposed model and obtained results can be used to guide the optimization design of multi-functional and multi-graded materials serviced under thermal environment.

On hygrothermal wave dispersion characteristics of embedded graphene foam

This investigation discusses wave propagation characteristics of porous graphene foam (PGF) beam resting on an elastic medium within the framework of the refined higher-order beam theory for the first time. The beam is considered to be in the hygrothermal environment. Additionally, the pores are distributed through the thickness direction uniformly and symmetrically. The motion equations of the PGF beam are derived via the Hamiltonian approach incorporated with the kinetic relation of refined higher-order shear deformation beam theory. Next, an analytical method is implemented in order to solve the governing equations of the PGF beam. The various parameters’ effect, including wave number, porosity coefficient, various types of porosity distribution, Winkler–Pasternak coefficients, hygrothermal environment, and slenderness ratio on the variation of wave frequency and phase velocity of PGF beam are investigated.

Natural frequency of Sandwich Beam Structures with Two Dimensional Functionally Graded Porous Layers Based on Novel Formulations

This study presents an analytical solution for free vibration analysis of two-dimensional functionally graded (2D-FG) porous sandwich beams. The equations of motion for the beam were derived using Hamilton's principle, and then the Galerkin method was employed to solve the equations. The material properties of the sandwich beams vary with the thickness and length of each layer according to the power-law function. The mechanical properties gradually changed from aluminum to alumina as the metal and ceramic, respectively. The vibration analysis was investigated based on two new higher-order shear deformation beam theories (NHSDBTs). These two new theories do not need any shear correction factor and have fewer unknown variables than other higher order shear beam theories. The obtained natural frequencies for the three types of beams were compared with the results of the Timoshenko, first-order , and parabolic shear deformation beam theories. In addition, the effects of porosity, L/h, and FG power indexes along the thickness and length on the non-dimensional frequency of three special types of beams are presented and discussed. Furthermore, the mode shapes of the beam are depicted for various FG power indexes based on these new theories. By comparing the results of the two proposed theories with those of existing studies, the accuracy of the proposed theories was validated. Power-law indexes shifted the node point to the left and resonance will be accrued sooner than the non-FGM beam.

Dynamic Analysis of a Rotating Double-Tapered FGM Beam with Various Shear Models Using a Constrained Variational Method

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively.

Thermal postbuckling analysis of sandwich beams with functionally graded auxetic GRMMC core on elastic foundations

This article deals with thermal postbuckling of sandwich beams caused by uniform temperature rise and resting on elastic foundations. The sandwich beam is composed of auxetic graphene-reinforced metal matrix composite (GRMMC) core with negative Poisson's ratio (NPR) and two metal face sheets. The volume fraction of graphene through the thickness domain of the GRMMC core is arranged as a functionally graded (FG) pattern. The thermo-mechanical properties of both metal face sheets and the GRMMC core are assumed to be temperature dependent. The governing equations of the auxetic sandwich beam are formulated based on the higher order shear deformation beam theory coupled with von Kármán-type kinematic nonlinearity. The thermal effects and the beam-foundation interaction are also taken into account in the modeling. Thermal postbuckling solutions are obtained by employing a two-step perturbation approach. The effects of the FG pattern, the face sheet-to-core-to-face sheet thickness ratio, and foundation stiffness on thermal postbuckling behavior of sandwich beams with FG auxetic GRMMC core are studied in detail.