Shear Deformation Beam Research Articles

A study on the crack presence effect on dynamical behaviour of bi-directional compositionally imperfect material graded micro beams

This work uses the Quasi-3D (Q3D) shear and normal deformation beam concept in combination with the length scale capturing theory through the “Modified Couple Stress Theory” (MCST) and the novel computational approach namely, Differential Quadrature Finite Element Method (DQFEM) to investigate free vibratory conducts of an imperfect BD-FG microbeams with a crack. To obtain the relevant governing equations, the Lagrange's rule is utilized. “The differential quadrature” (DQ) and “Gauss-Lobatto quadrature” procedures for weighting coefficient matrices are used to easily construct the mass and stiffness matrices. The Gauss-Lobatto node scheme is used to determine the griding knots. In order to show the efficiency and exactness of the described numerical resolution method, the attained outcomes are authenticated with the similar results found in the literature. The effects of the crack position, crack depth, beam thickness by material length scale aspect, the volume fraction of material imperfection, edge supports state types, length through thickness factor, and the two-way material grading indexes on the nondimensional natural frequencies of 2D porous FG microbeams are inspected. This work will provide a numerical basis for the design of FG microstructures in the field of micromechanics. These findings can be used to construct porous FG microstructures in engineering.

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

AbstractThe modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth‐order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second‐order coupled system into a first‐order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross‐sectional properties.

A Machine-Learning-Based Model for Buckling Analysis of Thermally Affected Covalently Functionalized Graphene/Epoxy Nanocomposite Beams

In this paper, a machine-learning model is utilized to estimate the temperature-dependent moduli of neat, thermally reduced graphene and covalently functionalized graphene/epoxy nanocomposites. In addition, the governed mathematical expressions have been used to solve the buckling problem of beams fabricated from such nanocomposites in the presence of a thermal gradient. In order to do so, an energy-based method including the shear deformable beam hypothesis is used. The beam structure is rested on the Winkler–Pasternak substrate. The reported verifications demonstrate the impressive precision of the presented ML model, as well as the buckling response of the under-study structures. Finally, in the framework of some numerical case studies, the impact of several parameters on the buckling of nanocomposite beams is depicted. The results of this study delineate that temperature has a vital role in the determination of the critical buckling load that the nanocomposite structures can endure.

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.

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.

Analytical solutions of Reddy, Timoshenko and Bernoulli beam models: A comparative analysis

The purpose of this paper is to compare the generalized stress/strain and corresponding local quantities obtained from three classical beam models commonly used in the literature: the Euler–Bernoulli, Timoshenko (or first-order shear deformation), and Reddy (or third-order shear deformation) beam models. In particular, we present an analytical solution for the equations governing the Reddy beam model, which is typically solved numerically. By expressing the solutions of all three models in terms of constants with the same physical meaning, we are able to make a direct comparison between them and identify the additional contributions of the higher-order models. We also compare the results with approximate solutions and find that while they provide satisfactory solutions for generalized displacement, they can lead to a significant underestimation of the stress field, with percentage errors that are greater than tolerable for practical purposes. This study provides new insights into the behavior of different beam models and their limitations in engineering applications.

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.

Vibration characteristics of arbitrary thick sandwich beam with metal foam core resting on elastic medium

An analysis is performed in this research to obtain the natural frequencies and mode shapes of a thick sandwich beam with metal foam core. Different types of foams are considered for the sandwich beam where distribution of pores is a functionally graded patterns. The governing equations of the sandwich beam are established by means of a shear and normal deformable thick beam model. This theory considers the thickness stretching and admits the nonuniform through-the-thickness shear strain. The governing equations of the sandwich beam are established which are four in number by means of the Hamilton principle. The established equations are solved for the case of thick sandwich beams resting on elastic foundation via the Navier solution method. Results of this study are first compared with the available data in the open literature and after that novel results are given to explore the effects of different parameters on the vibration characteristics of the sandwich beam. It is shown that porosity of the core is an important factor on vibration characteristics of sandwich beam with metal foam core.

Free vibrations of Timoshenko nanoscale beams based on a hybrid integral strain- and stress-driven nonlocal model

Several non-classical elasticity theories are used for considering the size-dependent behavior of structures at small scales. The nonlocal theory is widely used to reflect the softening behavior of material at small scales, and theories like the strain gradient theory are employed to reflect the hardening behavior. In this article, the most general form of integral strain- and stress-driven nonlocal models with two nonlocal parameters is developed which is able to consider both hardening and softening influences simultaneously. To this end, it is considered that the stress field at the entire points of the domain is a function of strain field of the entire points of the domain. The free vibration problem of first-order shear deformable beams is solved herein. The integral form of governing equations and associated boundary conditions are obtained first, and then directly solved in a numerical approach. Through developing an efficient matrix formulation and using differential and integral matrix operators, the discretized governing equations are obtained. The simultaneous effects of strain- and stress-driven nonlocal parameters on the natural frequencies of fully clamped, fully simply-supported, and clamped-free nanobeams are investigated. The results indicate that the paradox related to the behavior of clamped-free nanobeams is resolved using the presented integral nonlocal formulation. Also, it is revealed that it is possible to find some specific values of nonlocal parameters at which the prediction of hybrid nonlocal model coincides with that of classical elasticity theory.

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.

Dynamic finite element analysis of microbeams under a moving mass based on refined third-order shear deformation theory

The dynamic behavior of microbeams with rectangular cross section under action of a moving mass is studied in the present paper in the framework of a refined higher-order shear deformation beam theory. The modified couple stress theory (MCST) with only one additional scale parameter is adopted to describe the influence of the microsize effect on the dynamic response of the microbeams. A finite element formulation with ten degrees of freedom is formulated and used to establish the discretized equation of motion for the microbeams. The formulation, taking into account the influence of the inertial effect, the Coriolis and centrifugal forces resulted from the mass moving, is derived from the expressions of the elastic and kinetic energies of the microbeams. The accuracy and efficiency of the derived formulation are confirmed by comparing the result obtained in the present work with the published data. The dynamic response of the microbeams with simply supported ends, such as the curves for dimensionless mid-span deflection-moving time relationship, the dynamic magnification factors (DMFs) and the thickness distribution of stresses are assessed by using an implicit Newmark method. The obtained numerical results reveal that the material length scale parameter which is introduced in the MCST has an important role on the dynamic behavior of the microbeams, and the DMF obtained from the theory incoporating the MCST is considerably lower than that using the conventional beam theory. It is also shown that the amplitude of both the axial stress and shear stress is considerably decreased by the increase of the material length scale parameter. A numerical study is carried out to highlight the influence of various parameters such as the moving mass velocity and the the ratio of the total beam length to its height on the dynamic response of the microbeams.

Geometrically nonlinear analysis of sandwich composite beams reinforced by agglomeration carbon nanotubes

In this work, geometrically nonlinear behavior of sandwich composite beams reinforced by carbon nanotubes is studied, taking into account the influence of agglomeration of the carbon nanotubes (CNTs). The core of the sandwich beams is homogeneous while the two face sheets are made of CNT reinforced composite with the effective material properties being estimated by the Eshelby-Mori-Tanaka approach. A first-order shear deformable nonlinear beam element is formulated in the context of the total Lagrange formulation and used to construct the discretized nonlinear equilibrium equation. The Newton-Raphson based iterative procedure is used in conjunction with the arc-length method to trace the equilibrium paths of the beams. Detail parametric studies are carried out to illustrate the influence of the CNTs agglomeration, the amount of CNT volume fraction as well as the thicknesses of face sheets on the nonlinear behavior of the structure.

Analytical solution for free vibration analysis of GPL-RP beam integrated with piezoelectric layers

This report presents an analytical approach to the natural frequency analysis of a porous beam consisting of a host porous layer reinforced with graphene platelets (GPLs), namely GPL-reinforced porous core, and two piezoelectric outer layers. In the modelling, symmetric distributions of both porosity and GPLs in the core are supposed. The effective mechanical properties of the GPL-reinforced porous core are estimated by Halpin–Tsai model and the rule of mixture. The electric potential in each piezoelectric layer is assumed to vary linearly across its thickness. Two types of electrical boundary conditions, which are open- and closed-circuits, are considered for the free surfaces of the piezoelectric layers. Parabolic shear deformation beam theory associated with Hamilton’s principle is adopted to derive the governing equations of the free vibration. Afterwards these equations are solved analytically by Navier’s solution. Comparative and comprehensive studies are carried out to examine the accuracy and effects of parameters and conditions, such as GPL weight fraction, porosity coefficient, and electrical boundary conditions on the natural frequencies of the beam.

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.

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.

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

Nitsche-based isogeometric approach for free vibration analysis of laminated plate with multiple stiffeners and cutouts

On the basis of the first order shear deformation and Timoshenko beam theories, a general Nitsche-based isogeometric analysis (IGA) is developed for studying the free vibration behaviors of composite plates with/without stiffeners and cutouts. Firstly, the laminated plate with/without cutout is described by several non-overlapped NURBS patches. The stiffeners are placed on the common edges of adjacent patches and share the same control points with these sub-patch edges. Then, different patches are stuck together by the means of Nitsche method. Eventually, the proposed method is implemented for analyzing the free vibration of composite plates with/without stiffener and cutout. First of all, it avoids the discretization of the stiffener and the identification of plate elements where stiffener nodes are located at, that are required in finite element method and IGA. In addition, it can handle with laminated plate with multiple stiffeners and cutouts conveniently. At first, four meshing schemes with different element sizes are implemented to validate the good convergence and accuracy of the present method in case of laminated plate with stiffener. Then, rectangular, skew and elliptical plates without stiffener and cutout are considered and the obtained natural frequencies and mode shapes agree well with those reported in literature. Furthermore, results from composite plates with/without stiffener and cutout under different boundary conditions are analyzed by considering different lay-up numbers and orientations of fiber, shapes and numbers of cutout, orientation angles and numbers of stiffener.

Numerical Model for a Geometrically Nonlinear Analysis of Beams with Composite Cross-Sections

This paper presents a beam model for a geometrically nonlinear stability analysis of the composite beam-type structures. Each wall of the cross-section can be modeled with a different material. The nonlinear incremental procedure is based on an updated Lagrangian formulation where in each increment, the equilibrium equations are derived from the virtual work principle. The beam model accounts for the restrained warping and large rotation effects by including the nonlinear displacement field of the composite cross-section. First-order shear deformation theories for torsion and bending are included in the model through Timoshenko’s bending theory and a modified Vlasov’s torsion theory. The shear deformation coupling effects are included in the model using the six shear correction factors. The accuracy and reliability of the proposed numerical model are verified through a comparison of the shear-rigid and shear-deformable beam models in buckling problems. The obtained results indicated the importance of including the shear deformation effects at shorter beams and columns in which the difference that occurs is more than 10 percent.

Effects of initial compression/tension, foundation damping and pasternak medium on the dynamics of shear and normal deformable GPLRC beams under moving load

An attempt is made in this research to investigation the free vibration and dynamic response of an arbitrary thick composite beam which is reinforced with graphene platelets (GPLs). Distribution of GPLs in the layers may be different which leads to a functionally graded media. To this end a shear and normal deformable beam is adopted. The developed beam model has four unknowns and takes the non-uniform shear strains and also thickness stretching into account. Also the effects of compressive or tensile force, a two parameter Pasternak elastic foundation and also foundation damping are taken into account for both free and forced vibration responses. The governing equations of the beam are established using the Hamilton principle. By applying the Navier solution method suitable for simply supported edges, results of the present studies are provided. It should be noted that for forced vibration problem, Newmark time marching method is applied. For dynamic response the case of moving load is assumed. Results of this study are first compared with the available data in the open literature especially with those developed by 3D elasticity to show the accuracy of the developed formulation. After that novel results are given to discuss the effects of different parameters. It is highlighted that weight fraction and distribution pattern of GPLs are two important factors on the dynamic response of the structure.

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines, which reveals an interesting coarse mesh superconvergence. Firstly, the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived, where the degeneration to Timoshenko beams is discussed as well. Subsequently, in accordance with these frequency error measures, the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration. On the other hand, the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy. Meanwhile, it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates, in comparison with the ultimate fourth order accuracy as the meshes are progressively refined. Furthermore, the mesh size threshold for the coarse mesh superconvergence is provided as well. The proposed theoretical results are consistently proved by numerical experiments.