First-order Shear Deformation Beam Theory Research Articles

Active control for higher order micro sandwich beam with various cores integrated by piezoelectric layers based on MSGT on the Pasternak foundation

In the present article, the vibration suppression of higher order micro sandwich beams based on modified strain gradient beam theory (MSGT) and Reddy/Timoshenko beam theories is developed. Three types of core, including honeycomb, foam, and porous for a micro sandwich beam, are considered. The core’s top and bottom are integrated by piezoelectric layers as actuator and sensor, respectively. Displacement fields are defined via third-order shear deformation beam theory (TSDBT) (or Reddy beam theory (RBT)) and first-order shear deformation beam theory (FSDBT) (or Timoshenko beam theory (TBT)). The governing equations of motion are derived using MSGT to consider the small-scale effect. To optimal approaches for the controller, state feedback is designed based on a linear quadratic regulator (LQR) in the state-space form. Moreover, the effect of various parameters, including small-scale parameters, three types of core, temperature changes, Pasternak foundation, density of core, dimensionless cell thickness ([Formula: see text]), and internal aspect ratio ([Formula: see text]), angles ([Formula: see text]) of honeycomb, and the porosity parameter is performed on the active control of the smart micro sandwich beam. The numerical results indicate that the parameters mentioned significantly affect the optimal vibration control of the smart micro sandwich beam. It is found that the lightweight cores have more attenuation of the transient response and faster settling time. By analyzing the effect of temperature changes on active vibration control of lighter core, it was found that the frequency decreases as the temperature increases. Also, by considering the Pasternak foundation, it is found that the time period without considering the elastic foundation is longer than when considering the elastic foundation, and the natural frequency is vice versa. Also, the settling time by considering the elastic foundation occurs earlier. In the active control field of smart micro beams, this theoretical investigation developed widely as a benchmark for experimental research and the manufacturing process.

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.

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

On shear-dependent vibration of nano frames

In this study, the effect of shear deformation on the atomic size-dependent free vibration of nanoframes is investigated. The equations of motion of axial and bending vibrations are obtained via the variational algebra according to nonlocal elasticity theory. Axial and bending behaviors are expressed by using simple rod theory and first-order shear deformation beam theory, respectively. By combining the nonlocal finite element solutions for the equations of motion, an eigenvalue formulation for the free vibration of nanoframe is developed. In the presentation of numerical results, firstly, some comparison studies are carried out to show the accuracy of the present analysis. Then, the variations of nondimensional frequencies of three different nanoframe structures are investigated with respect to various parameters, and the results are discussed in detail. The focus of this study is to better understand the dynamic behavior of nanostructures that are within the scope of nano-electro-mechanical systems (NEMS) technology and can be modelled with discrete members.

Exact solution of thermal postbuckling of porous piezoelectric nanobeams with surface effect

In this paper, the thermal postbuckling of porous piezoelectric nanobeams is analyzed using the first-order shear deformation beam theory based on the surface elasticity theory and "core-shell" model, where the surface effect is introduced by employing the surface energy model. The governing equations for thermal postbuckling of porous piezoelectric nanobeams were derived by the principle of minimum potential energy. It is found that surface effect increases the effective elastic modulus and dimensionless critical load and reduces the thermal postbuckling conformations, thermal postbuckling paths and amount of induced charge. Meanwhile, the reasonable pore distribution pattern can effectively improve the mechanical properties.

Thermal buckling and postbuckling of functionally graded multilayer GPL-reinforced composite beams on nonlinear elastic foundations

Under the influence of axial forces and uniform temperature variations, the thermal buckling and postbuckling of composite beams reinforced of functionally graded multilayer graphene platelets (GPLs) resting on nonlinear elastic foundations are examined. The Halpin-Tsai model is used to calculate the elastic modulus of each layer of GPL-reinforced composite (GPLRC). According to the virtual work principle, the nonlinear governing equations for the beam are obtained from the first-order shear deformation beam theory. The impact of axial force and nonlinear elastic foundation on thermal buckling and postbuckling is discussed using the differential quadrature method (DQM), and the analytical expression is given by the two-step perturbation method (TSPM). The effects of axial force, boundary conditions, slenderness ratio, GPL geometry, GPL weight fraction, GPL distribution pattern, and elastic foundation coefficient on thermal buckling and postbuckling are examined through parameter analysis.

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

Vibrational characteristics of functionally graded graphene origami-enabled auxetic metamaterial beams based on machine learning assisted models

This paper studies the free vibration behavior and dynamic responses of functionally graded (FG) beams made of novel graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) under an impulsive load within the framework of the first-order shear deformation beam theory. The auxetic property of the beam is effectively controlled by graphene content and GOri folding degree that are graded across the thickness direction of the beams in a layer-wise manner such that Poisson's ratio and other material properties are position-dependent and are estimated by the genetic programming (GP)-assisted micromechanical models. The governing equations are derived by using Lagrange equation approach together with Ritz method. Newmark-β method is employed to solve the governing equations for obtaining dynamic responses of the beams subjected to three different impulsive loads. A comprehensive parametric study is performed to examine the effects of graphene content, GOri folding degree and distribution patterns as well as temperature on the natural frequencies and dynamic responses of the beams. Numerical results show that high tunability in structural vibration characteristics can be achieved via GOri, which offers important insight into the application of FG-GOEAM beams in aerospace engineering structure for significantly improved dynamic structural performance.

Geometrically nonlinear nonlocal strain gradient vibration of FG shear deformable curved nanobeams

This article focuses on studying the nonlinear vibration of functionally graded (FG) curved nanobeams resting on the Pasternak-Winkler elastic foundation based on the nonlocal strain gradient theory along with the first-order shear deformation beam theory (FSDBT) considering von-Kármán hypothesis. The Hamilton principle is applied to extract three nonlinear motion equations and the Galerkin method (GM) is utilized to spatially reduce the differential equations. The analytical approach based on the two-step perturbation method (TSPM) was employed to deal with nonlinear governing equations. To verify the outcomes of the present article, the natural frequencies and frequency ratios are validated with those reported in the literature. Subsequently, the results presented in this paper are of a significant point to describe the nonlinear vibration of FG curved nanobeams in conjunction with different parameters.

On the dynamic stability of delaminated composite beams under free vibration

This work deals with the analysis of the free vibration problem of elastic delaminated composite beams. The work mainly consists of a model development and improvement stage based on the first-order shear deformable beam theory. A general model is developed taking the bending-extensional coupling into account. The specified problem is a built-in beam with free end, and one of the novelties of this work is the consideration of the fact that a built-in beam cannot be fixed rigidly in reality. Thus, a Winkler-type elastic foundation is applied along the built-in length. The total potential energy and the governing equations of the delaminated and intact parts of the beam are also captured. The problem is solved in two ways: analytically and numerically by using the finite element method, respectively. Applying the developed models the natural frequencies, mode shapes as well as the stress resultants are determined. The comparison of natural frequencies to those measured experimentally shows that the built-in length resting on Winkler-type elastic foundation influences significantly the agreement between model and experiment. In the final stage, the parametric excitation phenomenon taking place in the delaminated part is analyzed using a local model and the harmonic balance method. The dynamic buckling is characterized by some stability diagrams, and it is shown that the applied model is very sensitive to the frequency leading to somewhat controversial critical amplitudes compared to measurements.

Low velocity impact study of a sandwich beams using Ritz method and finite element modelling

PurposeThe purpose of this paper is to present an applied method to design the low-speed contact between a mass and surface of a beam using an analytical solution based on the first-order shear deformation beam theory. Also, a simulation of impact process is carried out by ABAQUS finite element (FE) code.Design/methodology/approachIn theoretical formulation, first strains and stresses are obtained, then kinetic and potential energies are written, and using a combination of Ritz and Lagrange methods, a set of system of motion equations in the form of mass, stiffness and force matrices is obtained. Finally, the motion equations are solved using Runge–Kutta fourth order method.FindingsThe von Mises stress contours at the impact point and contact force from the ABAQUS simulation are illustrated and it is revealed that the theoretical solution is in good agreement with the FE code. The effect of changes in projectile speed, projectile diameter and projectile mass on the results is carefully examined with particular attention to evaluate histories of the impact force and beam recess. One of the important results is that changes in projectile speed have a greater effect on the results than changes in projectile diameter, and also changes in projectile mass have the least effect.Originality/valueThis paper presents a combination of methods of energy, Ritz and Lagrange and also FE code to simulate the problem of sandwich beams under low velocity impact.

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.

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.

Computer modeling via a numerical method for the cross-section effect on the forced vibration of FG nonuniform and imperfect microtubes

This paper examines the nonlinear forced vibrational behaviors of axially functionally graded (AFG) cylindrical microscale tubes under external dynamic stress, including the imperfect porosity. Using the Hamilton principle, the first-order shear deformation beam theory is linked with the modified couple stress theory (MCST) and the nonlinear von-Kármán theory to get the nonlinear governing equations and associated boundary conditions. For cross-sections that vary throughout the tube length, both uniform and non-uniform functions involving the three kinds are assumed. Furthermore, the microtube is constructed of functionally graded materials that vary smoothly throughout the tube length, and the porosity distribution is in the radial direction, resulting in two-dimensional functionally graded (2D-FG) structures with both porosity and material distributions. The nonlinear results are obtained after solving the linear motion equations using the generalized differential quadrature technique (GDQM) and the iteration approach. Finally, the effects of several factors on the natural and resonance frequencies are investigated, including the cross-section, FG parameter, porosity parameter, amplitude, external excitation frequency, and small-scale effect.

Multi-layer polymer beam reinforced by graphene platelets on low velocity impact response

PurposeThe purpose of this paper is to investigate the effect of graphene platelets (GPLs) on the low-speed contact between a mass and surface of a multi-layer polymer beam.Design/methodology/approachThis problem is primarily organized by first-order shear deformation beam theory and nonlinear Hertz rule. GPLs are distributed along the beam thickness direction. The Halpin–Tsai micromechanics model is applied for computing the effective Young’s modulus of the GPLs/polymer composites. In the formulation process, the principle of conservation of energy is first used and the histories of results are extracted using the separation of variables and Runge–Kutta method.FindingsIn comparing the responses with the available data, a good agreement is observed. The effects of the weight fraction and distribution pattern on the impact response of polymer beam reinforced with GPLs are studied. Results show that contact force is increased, contact time and beam recess are decreased with increasing of weight fraction of GPLs. Also, among the different distribution patterns, the contact force depended on value of GPLs at the point of contact.Originality/valueThe effects of GPLs addition on the multi-layer polymer beam has a novelty in impact problems.

Bending of Symmetric Sandwich FGM Beams with Shear Connectors

This paper carries out the static bending analysis of symmetric three-layer functionally graded sandwich beams, in which each layer is made from different functionally graded materials, and they are connected by shear connectors due to sliding movement. The finite element formulations are based on Timoshenko’s first-order shear deformation beam theory (FSDT) and the finite element method to establish the equilibrium equation of beams. The calculation program is coded in the MATLAB environment, and then verification examples are given out to compare the numerical data of present work with those of exact open sources. The impact of several geometrical and material parameters on the mechanical response of the structure, such as the height-to-length ratio, boundary conditions, volume fraction index, and especially the shear coefficient of connectors, is being explored. When designing and using these types of structures in engineering practice, the computed results can be utilized as a valid reference.

Forced Vibration Analysis of Composite Beams Reinforced by Carbon Nanotubes.

This paper presents forced vibration analysis of a simply supported beam made of carbon nanotube-reinforced composite material subjected to a harmonic point load at the midpoint of beam. The composite beam is made of a polymeric matrix and reinforced the single-walled carbon nanotubes with their various distributions. In the beam kinematics, the first-order shear deformation beam theory was used. The governing equations of problem were derived by using the Lagrange procedure. In the solution of the problem, the Ritz method was used, and algebraic polynomials were employed with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method was applied in the time history. In the numerical examples, the effects of carbon nanotube volume fraction, aspect ratio, and dynamic parameters on the forced vibration response of carbon nanotube-reinforced composite beams are investigated. In addition, some comparison studies were performed, with special results of published papers to validate the using formulations.

Bending analysis of functionally graded graphene oxide powder-reinforced composite beams using a meshfree method

In this paper, the linear bending response of straight, thin to moderately thick functionally graded (FG) composite beams reinforced with graphene oxide powder (GOP) and subjected to two types of static mechanical loads, with various boundary conditions is investigated using a meshfree collocation method based on multiquadric radial basis functions. The weight fractions of the GOP are considered to vary continuously through the thickness direction of the composite beams. To determine the effective Young's modulus of the FG-GOP reinforced composite beams, a modified Halpin–Tsai model is used, whereas the rule of mixture is adopted to estimate the equivalent Poisson's ratio. The first-order shear deformation (FSDT) beam theory is adopted to model the functionally graded graphene oxide powder (FG-GOP) reinforced composite beams. The static equilibrium equations and associated boundary conditions of the FG-GOP reinforced composite beams are obtained by using the principle of minimum potential energy. The governing equations are rewritten in a matrix-vector strong form and discretized using the multiquadric radial basis functions. Various comparisons studies are performed to demonstrate the robustness and accuracy of the proposed numerical model. Parametric studies are conducted to examine the effects of length-to-thickness ratio, GOP distribution patterns, weight fraction, and size of GOP, types of loads, and boundary conditions on the bending behavior of the FG-GOP reinforced composite beams.

Bending, buckling and free vibration behaviors of thin-walled functionally graded sandwich and composite channel-section beams

This article proposes static, free vibration, and buckling analysis of thin-walled functionally graded (FG) sandwich and composite channel-section beams. It is based on the first-order shear deformable beam theory, which can recover to classical one by ignoring the shear effect. Ritz’s approximation functions are developed to solve the characteristic problems. Both results from classical and the first-order shear deformable theories are given in a unified fashion. Ritz solutions are applied for thin-walled FG sandwich channel-section beams for the first time. Numerical examples are presented in relation to many important effects such as span-to-height ratio, material parameter, lay-ups, fiber orientation and boundary conditions on the beams’ deflections, natural frequencies, and critical buckling loads. New results presented in this study can be of interests to the scientific and engineering community in the future.