Higher-order Shear Deformation Beam Theory 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.

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.

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.

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.

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.

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.

Eringen’s nonlocal and modified couple stress theories applied to vibrating rotating nanobeams with temperature effects

This study develops a comprehensive vibrational analysis of rotating nanobeams on visco-elastic foundations with thermal effects based on the modified couple stress and Eringen’s nonlocal elasticity theories. This approach accurately simulates the nonlocal stress and size effects. Higher-order shear deformation beam theory and the generalized differential quadrature method are used to obtain the numerical results. The effects of nonlocal parameters, length scale, Winkler–Pasternak coefficients, thermal gradient, slenderness ratios, rotating velocity, and viscoelastic coefficient are demonstrated and discussed in detail. Mode switching and the importance of the correct choice of theory and associated size effect parameters are highlighted.

Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam

This work focused on the novel numerical tool for the bending responses of carbon nanotube reinforced composites (CNTRC) beams. The higher order shear deformation beam theory (HSDT) is used to determine strain-displacement relationships. A new exponential function was introduced into the carbon nanotube (CNT) volume fraction equation to show the effect of the CNT distribution on the CNTRC beams through displacements and stresses. To determine the mechanical properties of CNTRCs, the rule of the mixture was employed by assuming that the single-walled carbon nanotubes (SWCNTs)are aligned and distributed in the matrix. The governing equations were derived by Hamilton's principle, and the mathematical models presented in this work are numerically provided to verify the accuracy of the present theory. The effects of aspect ratio (l/d), CNT volume fraction (Vcnt), and the order of exponent (n) on the displacement and stresses are presented and discussed in detail. Based on the analytical results. It turns out that the increase of the exponent degree (n) makes the X-beam stiffer and the exponential CNTs distribution plays an indispensable role to improve the mechanical properties of the CNTRC beams.

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

Free vibration of radially graded hollow cylinders subject to axial force via a higher-order shear deformation beam theory

The dynamic behavior of radially functionally graded tubes is studied when subjected to axial tensile and compressive forces. Differing from the Euler–Bernoulli/Timoshenko beam theories, a higher-order shear beam deformation model that does neither require a shear correction factor nor need a planar cross-section assumption after deformation. The cross-section’s warping is constructed to meet the shear-free condition on tube’s inner and outer surfaces. A governing partial differential equation is derived. Exact characteristic equations for determining the natural frequencies are given. For typical end supports including hinged-hinged, clamped–clamped, clamped-free, and clamped-hinged ends, the natural frequencies are calculated. By letting the frequency vanish, the critical buckling loads under compression can be exactly determined by solving the characteristic equation. A comparison of the present buckling loads and the natural frequencies with the classical ones and with finite element results is made and verifies the efficiency of the proposed model. The frequency-load interaction is plotted for typical boundary conditions. Axial tensile force causes the natural frequencies to increase and axial compressive force decreases the natural frequencies. The effects of the radial gradient, tube’s thickness and length, and the cross-sectional warping shape on the buckling loads and the natural frequencies are elucidated.

Effect of Kerr Foundation and in-Plane Forces on Free Vibration of FGM Nanobeams with Diverse Distribution of Porosity

Abstract In the present paper, the effect of diverse distribution of functionally graded porous material and Kerr elastic foundation on natural vibrations of nanobeams subjected to in-plane forces is investigated based on the nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. All the displacements gradients are assumed to be small, then the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded (FG) porous material are expressed by nonlocal and length scale parameters and power-law variation of material parameters in conjunction with cosine functions. It created possibility to investigate an effect of functionally graded materials with diverse distribution of porosity and volume of voids on mechanics of structures in nano scale. The Hamilton’s variational principle is utilized to derive governing equations of motion of the FG porous nanobeam. Analytical solution to formulated boundary value problem is obtained in closed-form by using Navier solution technique. Validation of obtained results and parametric study are presented in tabular and graphical form. Influence of axial tensile/compressive forces and three different types of porosity distribution as well as stiffness of Kerr foundation on natural frequencies of functionally graded nanobeam is comprehensively studied.

Effect of warping shape on buckling of circular and rectangular columns under axial compression

The structural stability of a column with rectangular and circular cross-section under axial compression is studied based on various higher-order shear deformation beam theories. This paper has two-fold objectives. One is to introduce a transition parameter to describe the direction of axial load from Engesser’s hypothesis to Haringx’s one, then a unified method is presented to determine the critical load. The other is to introduce new cross-section warping shapes of rectangular and circular columns, then the buckling loads are exactly calculated and compared for various warping shapes. A governing equation for buckling of a column under axial compression is first derived. The buckling loads of a prismatic column with typical ends such as clamped-clamped, pinned-pinned, and clamped-free columns are determined and an explicit expression is obtained in terms of the Euler buckling load. The effects of the warping shape of the cross-section on the buckling loads are analyzed. The Haringx buckling load is greater than the Engesser buckling load that gives a conservative estimate of the critical load. A comparison of these buckling loads with Euler loads is made. The obtained results indicate that Euler buckling loads are both significantly overestimated for short columns or those with weak shear rigidity. The buckling loads are nearly not affected for very slender columns or those with high shear rigidity. The Euler loads are recovered from the present ones for columns in the case of shear locking. The buckling loads are also dependent on the warping shape of the cross-section.

Rational approach for higher-order shear deformation beam theories

The studies on the higher-order beam theory and its application to functionally graded (FG), laminated composite and sandwich beams have received much research attention over recent years. While significant progress has been made, there is an issue with the selection of the appropriate distribution function for describing section warping due to transverse loading. In the present study, a rational approach is proposed for the determination of correct warping functions for symmetric cross sections. Two new conditions on warping function are suggested: first, the warping function should vanish on the neutral plane; and second, the first derivative of warping function should take unity value on the neutral plane. A rigorous procedure is then developed by using the equilibrium condition and three conditions. An exact higher-order theory is then developed for creating accurate analysis models to consider both material and geometry variations over the beam cross section. A finite element analysis model is presented and a two-node beam element with bubble displacement modes is implemented. Results for sandwich and FG beam problems are shown and numerical results are compared with those of plane stress continuum models to demonstrate the correctness and application of the new theory and finite element model.