Shear Deformation Beam Theory Research Articles

Wave propagation and directionality in two-dimensional periodic lattices considering shear deformations

The effects of shear deformation on analysis of the wave propagation in periodic lattices are often assumed negligible. However, this assumption is not always true, especially for the lattices made of beams with smaller aspect ratios. Therefore, in the present paper, the effect of shear deformation on wave propagation in periodic lattices with different topologies is studied and their wave attenuation and directionality performances are compared. Current experimental limitations make the researchers focus more on the wave propagation in the direction perpendicular to the plane of periodicity in micro/nanoscale lattice materials while for their macro/mesoscale counterparts, in-plane modes can also be analyzed as well as the out-of-plane ones. Four well-known topologies of hexagonal, triangular, square, and Kagomé are considered in the current paper and their wave propagation is investigated both in the plane of periodicity and in the out-of-plane direction. The finite element method is used to formulate the governing equations and Bloch’s theorem is used to solve the dispersion relations. To investigate the effect of shear deformation, both the Timoshenko and Euler-Bernoulli beam theories are implemented. The results indicate that including shear deformation in wave propagation has a softening effect on the band diagrams of wave propagation and moves the dispersion branches to lower frequencies. It can also reveal some bandgaps that are not predicted without considering the shear deformation.

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.

Influence of porosity on the fundamental natural frequencies of FG sandwich beams

The concept of functionally graded materials (FGMs) is being actively investigated in sandwich beam design to enhance the durability of the sandwich structure. However, during the manufacturing of FGMs, some technical issues might cause porosities in FGMs, resulting in noticeable changes in mechanical characteristics of sandwich construction. Using an enhanced shear deformable beam theory, the influence of porosity on the fundamental natural frequencies of functionally graded (FG) sandwich beams having simply supported ends is examined in this study. To do so, the material properties of the FG sandwich beam (FGSB) are supposed to change incessantly in conjunction with the thickness, depending on the volume fraction of constituents specified by the modified rule of the mixture, which contains the volume fraction of porosity. The FGSB has two FG face sheets and a homogeneous hardcore (ceramic), and the FGSB has two FGface sheets and a homogeneous softcore (metal) are examined. The outcomes are reported for two constituent metal-ceramic FGSBs with a power-law change of the volume fractions of the constituents across the thickness. Consequently, it is seen that the volume fraction of porosity, power-law index, length to thickness ratio, and the thickness of the FG layer have a substantial impact on the dynamic response of FGSB according to the findings.

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.

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.

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.

Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory

The goal of this manuscript is to develop a nonclassical size dependent model to study and analyze the dynamic behaviour of the perforated Reddy nanobeam under moving load including the length scale and microstructure effects, that not considered before. The kinematic assumption of the third order shear deformation beam theory in conjunction with modified continuum constitutive equation of nonlocal strain gradient (NLSG) elasticity are proposed to derive the equation of motion of nanobeam included size scale (nonlocal) and microstructure (strain gradient) effects. Mathematical expressions for the equivalent geometrical parameters due to the perforation process of regular squared pattern are developed. Based on the virtual work principle, the governing equations of motion of perforated Reddy nanobeams are derived. Based on Navier's approach, an analytical solution procedure is developed to obtain free and forced vibration response under moving load. The developed methodology is verified and checked with previous works. Impacts of perforation, moving load velocity, microstructure parameter and nonlocal size scale effects on the dynamic and vibration responses of perforated Reddy nanobeam structures have been investigated in a wide context. The obtained results are supportive for the design of MEMS/NEMS structures such as frequency filters, resonators, relay switches, accelerometers, and mass flow sensors, with perforation.

On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory

In the present paper, a novel refined shear deformation beam theory is proposed and applied, for the first time, to investigate the bending behavior of functionally graded (FG) sandwich curved beam. The present theory is exploited to satisfy parabolic variation of shear stress distribution along the thickness direction thereby obviating the use of any shear correction factors. The material properties of FG sandwich beam change continuously from one surface to another according to a power-law function. Three common configurations of FG beams are considered for the study, namely: (a) single layer FG beam; (b) sandwich beam with FG face sheets and homogeneous core and (c) sandwich beams with homogeneous face sheets and FG core. The governing equations derived herein are solved by employing the finite element method using a two-noded beam element, developed for this purpose. A wide range of comparative studies is carried-out to establish the robustness and the performance of the present finite element model. A comparison study shows that the proposed model is: (a) accurate and comparable with the literature; b) of fast rate of convergence to the reference solution; c) excellent in terms of numerical stability and d) valid for both thin and thick FG sandwich curved beams. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, material distributions, length-to-thickness ratio, face-to-core- thickness ratio, loadings and boundary conditions on the static response of FG curved sandwich beam. New referential results for thin and thick FG sandwich curved beam are offered, in which can be used to establish benchmarks for future research.

Thermal effects on dynamic response of GOP-Reinforced beams under blast load

This article has been devoted to investigate the dynamic response of a graphene oxide powder (GOP) reinforced beam exposed to blast and thermal loads. It is assumed that the blast load has been exerted on the top surface of the beam. The material modeling of GOP-reinforced composite has been performed assuming two types of GOP distribution called uniform and graded in thickness direction. The establishment of the governing equations has been done employing a refined shear deformation beam theory and the consideration of elastic foundation. Finally, the equations have been solved implementing Chebyshev-Ritz method and inverse Laplace transform method. Then, the time history of the beam has been derived and its dependency on GOP percentage, GOP distribution, temperature rise, blast load factor, foundation coefficient band load position.

The effects of thermal loadings on wave propagation analysis of multi-scale hybrid composite beams

This research is concerned with the analysis of wave propagation of multi-scale hybrid composite beams subjected to various types of thermal loading. Carbon fiber (CF) and carbon nanotube (CNT) are assumed as reinforcements that are distributed in the matrix. The homogenization process is performed exerting the Halpin–Tsai model and the rule of mixture. Different types of temperature rise, namely, uniform, linear and sinusoidal temperature rise are presumed to present a more trustworthy thermal analysis. The beam is rested on Winkler–Pasternak foundation. A refined trigonometric shear deformable beam theory is used to compute the kinetic relations without any external shear correction coefficient. Governing equations are derived implementing Hamilton’s principle and then solved analytically. Eventually, the effects of different parameters, such as CNT’s weight fraction, the volume fraction of CF, elastic foundation, different types of temperature rise, etc., on wave frequency and phase velocity. are provided numerically in the framework of a set of illustrations.

Experimental, analytical and numerical investigation of pultruded GFRP composite beams infilled with hybrid FRP reinforced concrete

This paper presents the experimental, analytical and numerical analyses on the behavior of the pultruded GFRP composite beams infilled with hybrid fiber reinforced concrete under four-point loading. The examined experimental variables included: (1) effect of pultruded GFRP box profile, (2) effect of conventional steel bars, GFRP bars and hybrid bars, (3) effect of externally GFRP wrapping. Pursuant to this goal, a total of 9 medium-sized beam specimens were constructed. The experimental tests were simulated using finite element models with ABAQUS program. A novel analytical method for analyzing the flexure behavior of beams has been presented. Applying the first order shear deformation beam theory and introducing auxiliary functions, the equation of motion is derived using the Hamiltonian approach. Initial stiffness, ductility, energy dissipation capacity were compared based on the load–displacement relationship. The pultruded profile significantly enhanced the behavior of traditional reinforced concrete beam and GFRP composite wrapping also considerably improved the behavior of the pultruded profile infilled with reinforced concrete beam. More importantly is that hybrid reinforcements are recommended for all cases. Moreover, detailed damage analyses are provided.

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.

Nonlinear dynamics of two-directional functionally graded microbeam with geometrical imperfection using unified shear deformable beam theory

In this paper, the nonlinear dynamic response of two-directional functionally graded (2D-FG) microbeam incorporating geometrically imperfect effect is investigated via a unified shear deformable beam theory, which can degenerate into several previous theories including Euler-Bernoulli, Timoshenko, and Reddy shear deformable beam theories. The material properties of 2D-FG microbeam are assumed to be varied continually along both the axial and thickness directions following the power-law distribution, and two patterns of material distribution along the thickness direction are considered. Both the global and localized imperfection modes are taken into account by the product of trigonometric and hyperbolic functions. Employing modified couple stress theory and the Hamilton's principle, the governing equations of the imperfect 2D-FG microbeam are derived and solved with the aid of the differential quadrature method. The influences of material distribution pattern, geometrical imperfection shape and amplitude, axial and thickness power-law indices, as well as length scale parameter and boundary condition on the nonlinear vibration performance of the 2D-FG microbeam are examined in detail. It is expected that the presented numerical results can be used to guide the optimal design of multi-functional micro-structures.

Dynamic analysis in beam element of wave-piercing Catamarans undergoing slamming load based on mathematical modelling

Dynamic analysis of wave-piercing Catamarans undergoing slamming has tracked widespread attention in the design of marine structures. Hence, our investigation on the dynamic response relies heavily on the beam element of wave-piercing Catamarans subjected to slamming load. In the present paper, a numerical method and a mathematical modelling are developed to analyze the dynamic response of beam elements in wave-piercing Catamarans due to impacting moving fluid load. With assuming the initial phase of slamming force, an elastic linear hyperbolic shear deformation beam theory (HSDBT) with uniform thickness is utilized for modelling. For presenting a more realistic model, the porosity and structural damping of the Aluminum beam are considered. In addition, for improving the stiffness of the mentioned structure against the slamming load, graphene platelet (GPLs) are used for reinforcement of the beam. To capture the dynamic deflection of the porous GPL-reinforced beam, differential quadrature method (DQM) along with Newmark method is utilized. The aim of this work is to illustrate the influences of deadrise angle, vertical speed of impact, GPLs volume percent and distribution, geometrical parameters, porosity, structural damping and boundary conditions on the dynamic deflection of the structure. The results indicated that reinforcing the facesheets by CNTs reduces the dynamic deflection up to 59%.

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.

Dynamic stability analysis of delaminated composite beams in frequency domain using a unified beam theory with higher order displacement continuity

In this paper the dynamic stability behaviour of single delaminated composite beam is investigated using higher order shear deformable beam theories. The system is discretised by the finite element method and the mass, material stiffness and geometric stiffness matrices are formulated by means of higher order displacement continuity in the intact part. The natural frequencies and critical buckling forces are determined for various beam theory orders and delamination lengths and positions. Furthermore, using the multi frequency method the dynamic stability properties of the system are calculated in frequency domain. Based on the stability analysis the sufficient beam theory order and multi frequency method’s corresponding truncated matrix determinant order are determined in order to achieve the desired accuracy on the observed parameter plane. The results indicate that the dependence of the natural frequencies and buckling forces is negligible on the order of the applied beam theory. In contrast the former quantities are significantly influenced by the delamination length and position. Finally, it is shown that the dynamic stability diagrams and the boundary frequencies of parametric instability do not depend on the delamination length and position from the engineering point of view.

Examination of thermal postbuckling of temperature dependent FG-GRMMC laminated beams with negative Poisson’s ratio on elastic foundations

This paper examines the in-plane negative Poisson’s ratio (NPR) effect on the thermal postbuckling of graphene-reinforced metal matrix composite (GRMMC) laminated beams subjected to a uniform temperature rise and supported by an elastic foundation. GRMMC layers having different graphene volume fractions can facilitate the formation of functionally graded (FG) patterns of the beams. The temperature-dependent material properties of GRMMC layers can be estimated by a nonlinear function of temperature from the results of the molecular dynamics simulations. The governing equations for the beam thermal postbuckling problem can be derived based on a higher order shear deformation beam theory and solved by applying the two-step perturbation approach. The von Kármán geometric nonlinearity, the elastic foundation support and the thermal effect are considered in the modeling. Numerical investigations are carried out for (10/-10/10/-10/10)S and (10/-10/10)S GRMMC beams with in-plane NPR. The results show that the buckling temperature of (10/-10/10)S UD beam is slightly higher than that of (10/-10/10/-10/10)S UD beam. The thermal postbuckling capacity for the (10/-10/10)S FG-X beam becomes much higher than that of the (10/-10/10/-10/10)S FG-X beam, when the beam deflection is sufficiently large.

Bending analysis of functionally graded porous nanocomposite beams based on a non-local strain gradient theory

In the present work we study the static response of functionally graded (FG) porous nanocomposite beams, with a uniform or non-uniform layer-wise distribution of the internal pores and graphene platelets (GPLs) reinforcing phase in the matrix, according to three different patterns. The finite-element approach is developed here together with a non-local strain gradient theory and a novel trigonometric two-variable shear deformation beam theory, to study the combined effects of the non-local stress and strain gradient on the FG structure. The governing equations of the problem are solved introducing a three-node beam element. A comprehensive parametric study is carried out on the bending behavior of nanocomposite beams, with a particular focus on their sensitivity to the weight fraction and distribution pattern of GPLs reinforcement, as well as to the non-local scale parameters, geometrical properties, and boundary conditions. Based on the results, it seems that the porosity distribution and GPLs pattern have a meaningful effect on the structural behavior of nanocomposite beams, where the optimal response is reached for a non-uniform and symmetric porosity distribution and GPLs dispersion pattern within the material.